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15 Venn Diagram Questions And Practice Problems (Middle & High School): Exam Style Questions Included
Beki Christian
Venn diagram questions involve visual representations of the relationship between two or more different groups of things. Venn diagrams are first covered in elementary school and their complexity and uses progress through middle and high school.
This article will look at the types of Venn diagram questions that might be encountered at middle school and high school, with a focus on exam style example questions and preparing for standardized tests. We will also cover problemsolving questions. Each question is followed by a worked solution.
How to solve Venn diagram questions
In middle school, sets and set notation are introduced when working with Venn diagrams. A set is a collection of objects. We identify a set using braces. For example, if set A contains the odd numbers between 1 and 10, then we can write this as:
A = {1, 3, 5, 7, 9}
Venn diagrams sort objects, called elements, into two or more sets.
This diagram shows the set of elements
{1,2,3,4,5,6,7,8,9,10} sorted into the following sets.
Set A= factors of 10
Set B= even numbers
The numbers in the overlap (intersection) belong to both sets. Those that are not in set A or set B are shown outside of the circles.
Different sections of a Venn diagram are denoted in different ways.
ξ represents the whole set, called the universal set.
∅ represents the empty set, a set containing no elements.
Venn Diagrams Worksheet
Download this quiz to check your students' understanding of Venn diagrams. Includes 10 questions with answers!
Let’s check out some other set notation examples!
A \cap B \quad  ^{\prime}\mathrm{A} and B^{\prime} The intersection of \mathrm{A} and \mathrm{B} . The elements in both sets \mathrm{A} and \mathrm{B.} \quad  
A \cup B \quad  ^{\prime}\mathrm{A} or B^{\prime} The union of \mathrm{A} or \mathrm{B.} . Any element in set \mathrm{A} or set \mathrm{B.}  
A^{\prime}  ‘Not \mathrm{A}^{\prime} The complement of \mathrm{A.} Any element not in \mathrm{A.} 
In middle school and high school, we often use Venn diagrams to establish probabilities.
We do this by reading information from the Venn diagram and applying the following formula.
For Venn diagrams we can say
Middle School Venn diagram questions
In middle school, students learn to use set notation with Venn diagrams and start to find probabilities using Venn diagrams. The questions below are examples of questions that students may encounter in 6th, 7th and 8th grade.
Venn diagram questions 6th grade
1. This Venn diagram shows information about the number of people who have brown hair and the number of people who wear glasses.
How many people have brown hair and glasses?
The intersection, where the Venn diagrams overlap, is the part of the Venn diagram which represents brown hair AND glasses. There are 4 people in the intersection.
2. Which set of objects is represented by the Venn diagram below?
We can see from the Venn diagram that there are two green triangles, one triangle that is not green, three green shapes that are not triangles and two shapes that are not green or triangles. These shapes belong to set D.
Venn diagram questions 7th grade
3. Max asks 40 people whether they own a cat or a dog. 17 people own a dog, 14 people own a cat and 7 people own a cat and a dog. Choose the correct representation of this information on a Venn diagram.
There are 7 people who own a cat and a dog. Therefore, there must be 7 more people who own a cat, to make a total of 14 who own a cat, and 10 more people who own a dog, to make a total of 17 who own a dog.
Once we put this information on the Venn diagram, we can see that there are 7+7+10=24 people who own a cat, a dog or both.
4024=16 , so there are 16 people who own neither.
4. The following Venn diagrams each show two sets, set A and set B . On which Venn diagram has A ′ been shaded?
\mathrm{A}^{\prime} means not in \mathrm{A} . This is shown in diagram \mathrm{B.}
Venn diagram questions 8th grade
5. Place these values onto the following Venn diagram and use your diagram to find the number of elements in the set \text{S} \cup \text{O}.
\xi = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \text{S} = square numbers \text{O} = odd numbers
\text{S} \cup \text{O} is the union of \text{S} or \text{O} , so it includes any element in \text{S} , \text{O} or both. The total number of elements in \text{S} , \text{O} or both is 6.
6. The Venn diagram below shows a set of numbers that have been sorted into prime numbers and even numbers.
A number is chosen at random. Find the probability that the number is prime and not even.
The section of the Venn diagram representing prime and not even is shown below.
There are 3 numbers in the relevant section out of a possible 10 numbers altogether. The probability, as a fraction, is \frac{3}{10}.
7. Some people visit the theater. The Venn diagram shows the number of people who bought ice cream and drinks in the interval.
Ice cream is sold for $3 and drinks are sold for $ 2. A total of £262 is spent. How many people bought both a drink and an ice cream?
Money spent on drinks: 32 \times \$2 = \$64
Money spent on ice cream: 16 \times \$3 = \$48
\$64+\$48=\$112 , so the information already on the Venn diagram represents \$112 worth of sales.
\$262\$112 = \$150 , so another \$150 has been spent.
If someone bought a drink and an ice cream, they would have spent \$2+\$3 = \$5.
\$150 \div \$5=30 , so 30 people bought a drink and an ice cream.
High school Venn diagram questions
In high school, students are expected to be able to take information from word problems and put it onto a Venn diagram involving two or three sets. The use of set notation is extended and the probabilities become more complex.
In advanced math classes, Venn diagrams are used to calculate conditional probability.
Lower ability Venn diagram questions
8. 50 people are asked whether they have been to France or Spain.
18 people have been to France. 23 people have been to Spain. 6 people have been to both.
By representing this information on a Venn diagram, find the probability that a person chosen at random has not been to Spain or France.
6 people have been to both France and Spain. This means 17 more have been to Spain to make 23 altogether, and 12 more have been to France to make 18 altogether. This makes 35 who have been to France, Spain or both and therefore 15 who have been to neither.
The probability that a person chosen at random has not been to France or Spain is \frac{15}{50}.
9. Some people were asked whether they like running, cycling or swimming. The results are shown in the Venn diagram below.
One person is chosen at random. What is the probability that the person likes running and cycling?
9 people like running and cycling (we include those who also like swimming) out of 80 people altogether. The probability that a person chosen at random likes running and cycling is \frac{9}{80}.
10. ξ = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16\}
\text{A} = \{ even numbers \}
\text{B} = \{ multiples of 3 \}
By completing the following Venn diagram, find \text{P}(\text{A} \cup \text{B}^{\prime}).
\text{A} \cup \text{B}^{\prime} means \text{A} or not \text{B} . We need to include everything that is in \text{A} or is not in \text{B} . There are 13 elements in \text{A} or not in \text{B} out of a total of 16 elements.
Therefore \text{P}(\text{A} \cup \text{B}^{\prime}) = \frac{13}{16}.
11. ξ = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}
A = \{ multiples of 2 \}
By putting this information onto the following Venn diagram, list all the elements of B.
We can start by placing the elements in \text{A} \cap \text{B} , which is the intersection.
We can then add any other multiples of 2 to set \text{A}.
Next, we can add any unused elements from \text{A} \cup \text{B} to \text{B}.
Finally, any other elements can be added to the outside of the Venn diagram.
The elements of \text{B} are \{1, 2, 3, 4, 6, 12\}.
Middle ability high school Venn diagram questions
12. Some people were asked whether they like strawberry ice cream or chocolate ice cream. 82% said they like strawberry ice cream and 70% said they like chocolate ice cream. 4% said they like neither.
By putting this information onto a Venn diagram, find the percentage of people who like both strawberry and chocolate ice cream.
Here, the percentages add up to 156\%. This is 56\% too much. In this total, those who like chocolate and strawberry have been counted twice and so 56\% is equal to the number who like both chocolate and strawberry. We can place 56\% in the intersection, \text{C} \cap \text{S}
We know that the total percentage who like chocolate is 70\%, so 7056 = 14\%14\% like just chocolate. Similarly, 82\% like strawberry, so 8256 = 26\%26\% like just strawberry.
13. The Venn diagram below shows some information about the height and gender of 40 students.
A student is chosen at random. Find the probability that the student is female given that they are over 1.2 m .
We are told the student is over 1.2m. There are 20 students who are over 1.2m and 9 of them are female. Therefore the probability that the student is female given they are over 1.2m is \frac{9}{20}.
14. The Venn diagram below shows information about the number of students who study history and geography.
H = history
G = geography
Work out the probability that a student chosen at random studies only history.
We are told that there are 100 students in total. Therefore:
x = 12 or x = 3 (not valid) If x = 12, then the number of students who study only history is 12, and the number who study only geography is 24. The probability that a student chosen at random studies only history is \frac{12}{100}.
15. 50 people were asked whether they like camping, holiday home or hotel holidays.
18\% of people said they like all three. 7 like camping and holiday homes but not hotels. 11 like camping and hotels. \frac{13}{25} like camping.
Of the 27 who like holiday homes, all but 1 like at least one other type of holiday. 7 people do not like any of these types of holiday.
By representing this information on a Venn diagram, find the probability that a person chosen at random likes hotels given that they like holiday homes.
Put this information onto a Venn diagram.
We are told that the person likes holiday homes. There are 27 people who like holiday homes. 19 of these also like hotels. Therefore, the probability that the person likes hotels given that they like holiday homes is \frac{19}{27}.
Looking for more Venn diagram math questions for middle and high school students ?
 Probability questions
 Ratio questions
 Algebra questions
 Trigonometry questions
 Long division questions
 Pythagorean theorem questions
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The content in this article was originally written by secondary teacher Beki Christian and has since been revised and adapted for US schools by elementary math teacher Katie Keeton.
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Venn Diagram Examples, Problems and Solutions
On this page:
 What is Venn diagram? Definition and meaning.
 Venn diagram formula with an explanation.
 Examples of 2 and 3 sets Venn diagrams: practice problems with solutions, questions, and answers.
 Simple 4 circles Venn diagram with word problems.
 Compare and contrast Venn diagram example.
Let’s define it:
A Venn Diagram is an illustration that shows logical relationships between two or more sets (grouping items). Venn diagram uses circles (both overlapping and nonoverlapping) or other shapes.
Commonly, Venn diagrams show how given items are similar and different.
Despite Venn diagram with 2 or 3 circles are the most common type, there are also many diagrams with a larger number of circles (5,6,7,8,10…). Theoretically, they can have unlimited circles.
Venn Diagram General Formula
n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
Don’t worry, there is no need to remember this formula, once you grasp the meaning. Let’s see the explanation with an example.
This is a very simple Venn diagram example that shows the relationship between two overlapping sets X, Y.
X – the number of items that belong to set A Y – the number of items that belong to set B Z – the number of items that belong to set A and B both
From the above Venn diagram, it is quite clear that
n(A) = x + z n(B) = y + z n(A ∩ B) = z n(A ∪ B) = x +y+ z.
Now, let’s move forward and think about Venn Diagrams with 3 circles.
Following the same logic, we can write the formula for 3 circles Venn diagram :
n(A ∪ B ∪ C) = n(A) + n(B) + n(C) – n(A ∩ B) – n(B ∩ C) – n(C ∩ A) + n(A ∩ B ∩ C)
Venn Diagram Examples (Problems with Solutions)
As we already know how the Venn diagram works, we are going to give some practical examples (problems with solutions) from the real life.
2 Circle Venn Diagram Examples (word problems):
Suppose that in a town, 800 people are selected by random types of sampling methods . 280 go to work by car only, 220 go to work by bicycle only and 140 use both ways – sometimes go with a car and sometimes with a bicycle.
Here are some important questions we will find the answers:
 How many people go to work by car only?
 How many people go to work by bicycle only?
 How many people go by neither car nor bicycle?
 How many people use at least one of both transportation types?
 How many people use only one of car or bicycle?
The following Venn diagram represents the data above:
Now, we are going to answer our questions:
 Number of people who go to work by car only = 280
 Number of people who go to work by bicycle only = 220
 Number of people who go by neither car nor bicycle = 160
 Number of people who use at least one of both transportation types = n(only car) + n(only bicycle) + n(both car and bicycle) = 280 + 220 + 140 = 640
 Number of people who use only one of car or bicycle = 280 + 220 = 500
Note: The number of people who go by neither car nor bicycle (160) is illustrated outside of the circles. It is a common practice the number of items that belong to none of the studied sets, to be illustrated outside of the diagram circles.
We will deep further with a more complicated triple Venn diagram example.
3 Circle Venn Diagram Examples:
For the purposes of a marketing research , a survey of 1000 women is conducted in a town. The results show that 52 % liked watching comedies, 45% liked watching fantasy movies and 60% liked watching romantic movies. In addition, 25% liked watching comedy and fantasy both, 28% liked watching romantic and fantasy both and 30% liked watching comedy and romantic movies both. 6% liked watching none of these movie genres.
Here are our questions we should find the answer:
 How many women like watching all the three movie genres?
 Find the number of women who like watching only one of the three genres.
 Find the number of women who like watching at least two of the given genres.
Let’s represent the data above in a more digestible way using the Venn diagram formula elements:
 n(C) = percentage of women who like watching comedy = 52%
 n(F ) = percentage of women who like watching fantasy = 45%
 n(R) = percentage of women who like watching romantic movies= 60%
 n(C∩F) = 25%; n(F∩R) = 28%; n(C∩R) = 30%
 Since 6% like watching none of the given genres so, n (C ∪ F ∪ R) = 94%.
Now, we are going to apply the Venn diagram formula for 3 circles.
94% = 52% + 45% + 60% – 25% – 28% – 30% + n (C ∩ F ∩ R)
Solving this simple math equation, lead us to:
n (C ∩ F ∩ R) = 20%
It is a great time to make our Venn diagram related to the above situation (problem):
See, the Venn diagram makes our situation much more clear!
From the Venn diagram example, we can answer our questions with ease.
 The number of women who like watching all the three genres = 20% of 1000 = 200.
 Number of women who like watching only one of the three genres = (17% + 12% + 22%) of 1000 = 510
 The number of women who like watching at least two of the given genres = (number of women who like watching only two of the genres) +(number of women who like watching all the three genres) = (10 + 5 + 8 + 20)% i.e. 43% of 1000 = 430.
As we mentioned above 2 and 3 circle diagrams are much more common for problemsolving in many areas such as business, statistics, data science and etc. However, 4 circle Venn diagram also has its place.
4 Circles Venn Diagram Example:
A set of students were asked to tell which sports they played in school.
The options are: Football, Hockey, Basketball, and Netball.
Here is the list of the results:
Football  Robert, James, John, Mary, Jennifer, William 
Hockey  Robert, William, Linda, Elizabeth, James 
Basketball  William, Jayne, Linda, Daniel, Mary 
Netball  Jessica, William, Linda, Elizabeth, Anthony, Mary 
None  Dorothy 
The next step is to draw a Venn diagram to show the data sets we have.
It is very clear who plays which sports. As you see the diagram also include the student who does not play any sports (Dorothy) by putting her name outside of the 4 circles.
From the above Venn diagram examples, it is obvious that this graphical tool can help you a lot in representing a variety of data sets. Venn diagram also is among the most popular types of graphs for identifying similarities and differences .
Compare and Contrast Venn Diagram Example:
The following compare and contrast example of Venn diagram compares the features of birds and bats:
Tools for creating Venn diagrams
It is quite easy to create Venn diagrams, especially when you have the right tool. Nowadays, one of the most popular way to create them is with the help of paid or free graphing software tools such as:
You can use Microsoft products such as:
Some free mind mapping tools are also a good solution. Finally, you can simply use a sheet of paper or a whiteboard.
Conclusion:
A Venn diagram is a simple but powerful way to represent the relationships between datasets. It makes understanding math, different types of data analysis , set theory and business information easier and more fun for you.
Besides of using Venn diagram examples for problemsolving and comparing, you can use them to present passion, talent, feelings, funny moments and etc.
Be it data science or realworld situations, Venn diagrams are a great weapon in your hand to deal with almost any kind of information.
If you need more chart examples, our posts fishbone diagram examples and what does scatter plot show might be of help.
About The Author
Silvia Valcheva
Silvia Valcheva is a digital marketer with over a decade of experience creating content for the tech industry. She has a strong passion for writing about emerging software and technologies such as big data, AI (Artificial Intelligence), IoT (Internet of Things), process automation, etc.
Well explained I hope more on this one
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Venn Diagram Examples for Problem Solving
Updated on: 11 September 2024
What is a Venn Diagram?
Venn diagrams define all the possible relationships between collections of sets. The most basic Venn diagrams simply consist of multiple circular boundaries describing the range of sets.
The overlapping areas between the two boundaries describe the elements which are common between the two, while the areas that aren’t overlapping house the elements that are different. Venn diagrams are used often in math that people tend to assume they are used only to solve math problems. But as the 3 circle Venn diagram below shows it can be used to solve many other problems.
Though the above diagram may look complicated, it is actually very easy to understand. Although Venn diagrams can look complex when solving business processes understanding of the meaning of the boundaries and what they stand for can simplify the process to a great extent. Let us have a look at a few examples which demonstrate how Venn diagrams can make problem solving much easier.
Example 1: Company’s Hiring Process
The first Venn diagram example demonstrates a company’s employee shortlisting process. The Human Resources department looks for several factors when shortlisting candidates for a position, such as experience, professional skills and leadership competence. Now, all of these qualities are different from each other, and may or may not be present in some candidates. However, the best candidates would be those that would have all of these qualities combined.
The candidate who has all three qualities is the perfect match for your organization. So by using simple Venn Diagrams like the one above, a company can easily demonstrate its hiring processes and make the selection process much easier.
A colorful and precise Venn diagram like the above can be easily created using our Venn diagram maker and we have professionally designed Venn diagram templates for you to get started fast too.
Example 2: Investing in a Location
The second Venn diagram example takes things a step further and takes a look at how a company can use a Venn diagram to decide a suitable office location. The decision will be based on economic, social and environmental factors.
In a perfect scenario you’ll find a location that has all the above factors in equal measure. But if you fail to find such a location then you can decide which factor is most important to you. Whatever the priority because you already have listed down the locations making the decision becomes easier.
Example 3: Choosing a Dream Job
The last example will reflect on how one of the life’s most complicated questions can be easily answered using a Venn diagram. Choosing a dream job is something that has stumped most college graduates, but with a single Venn diagram, this thought process can be simplified to a great extent.
First, single out the factors which matter in choosing a dream job, such as things that you love to do, things you’re good at, and finally, earning potential. Though most of us dream of being a celebrity and coming on TV, not everyone is gifted with acting skills, and that career path may not be the most viable. Instead, choosing something that you are good at, that you love to do along with something that has a good earning potential would be the most practical choice.
A job which includes all of these three criteria would, therefore, be the dream job for someone. The three criteria need not necessarily be the same, and can be changed according to the individual’s requirements.
So you see, even the most complicated processes can be simplified by using these simple Venn diagrams.
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Great article, and all true, but.. I hate venn diagrams! I don’t know why, they’ve just never seemed to work for me. Frustrating!
Hey thanks for writing. It helped me in many ways Thanks again 🙂
Hi Nishadha,
Nice article! I love Venn Diagrams because nothing comes to close to expressing the logical relationships between different sets of elements that well. With Microsoft Word 2003 you can create fantastic looking and colorful Venn Diagrams on the fly, with as many elements and colors as you need.
Hi Worli, Yes, Venn diagrams are a good way to solve problems, it’s a shame that it’s sort of restricted to the mathematics subject. MS Word do provides some nice options to create Venn diagrams, although it’s not the cheapest thing around.
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Venn Diagram
A Venn diagram is used to visually represent the differences and the similarities between two concepts. Venn diagrams are also called logic or set diagrams and are widely used in set theory, logic, mathematics, businesses, teaching, computer science, and statistics.
Let's learn about Venn diagrams, their definition, symbols, and types with solved examples.
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What is a Venn Diagram?
A Venn diagram is a diagram that helps us visualize the logical relationship between sets and their elements and helps us solve examples based on these sets. A Venn diagram typically uses intersecting and nonintersecting circles (although other closed figures like squares may be used) to denote the relationship between sets.
Venn Diagram Example
Let us observe a Venn diagram example. Here is the Venn diagram that shows the correlation between the following set of numbers.
 One set contains even numbers from 1 to 25 and the other set contains the numbers in the 5x table from 1 to 25.
 The intersecting part shows that 10 and 20 are both even numbers and also multiples of 5 between 1 to 25.
Terms Related to Venn Diagram
Let us understand the following terms and concepts related to Venn Diagram, to understand it better.
Universal Set
Whenever we use a set, it is easier to first consider a larger set called a universal set that contains all of the elements in all of the sets that are being considered. Whenever we draw a Venn diagram:
 A large rectangle is used to represent the universal set and it is usually denoted by the symbol E or sometimes U.
 All the other sets are represented by circles or closed figures within this larger rectangle .
 Every set is the subset of the universal set U.
Consider the abovegiven image:
 U is the universal set with all the numbers 110, enclosed within the rectangle.
 A is the set of even numbers 110, which is the subset of the universal set U and it is placed inside the rectangle.
 All the numbers between 110, that are not even, will be placed outside the circle and within the rectangle as shown above.
Venn diagrams are used to show subsets. A subset is actually a set that is contained within another set. Let us consider the examples of two sets A and B in the belowgiven figure. Here, A is a subset of B. Circle A is contained entirely within circle B. Also, all the elements of A are elements of set B.
This relationship is symbolically represented as A ⊆ B. It is read as A is a subset of B or A subset B. Every set is a subset of itself. i.e. A ⊆ A. Here is another example of subsets :
 N = set of natural numbers
 I = set of integers
 Here N ⊂ I, because allnatural numbers are integers .
Venn Diagram Symbols
There are more than 30 Venn diagram symbols. We will learn about the three most commonly used symbols in this section. They are listed below as:
Venn Diagram Symbols  Explanation 

The union symbol  ∪  A ∪ B is read as A union B. Elements that belong to either set A or set B or both the sets. U is the universal set. 
The intersection symbol  ∩  A ∩ B is read as A intersection B. Elements that belong to both sets A and B. U is the universal set. 
The complement symbol  A or A'  A' is read as A complement. Elements that don't belong to set A. U is the universal set. 
Let us understand the concept and the usage of the three basic Venn diagram symbols using the image given below.
Symbol  It refers to  Total Elements (No. of students) 

A ∪ C  The number of students that prefer either burger or pizza or both.  1 + 10 + 2 + 2 + 6 + 9 = 30 
A ∩ C  The number of students that prefer both burger and pizza.  2 + 2 = 4 
A ∩ B ∩ C  The number of students that prefer a burger, pizza as well as hotdog.  2 
A or A'  The number of students that do not prefer a burger.  10 + 6 + 9 = 25 
Venn Diagram for Sets Operations
In set theory, we can perform certain operations on given sets. These operations are as follows,
 Union of Set
 Intersection of set
 Complement of set
 Difference of set
Union of Sets Venn Diagram
The union of two sets A and B can be given by: A ∪ B = {x  x ∈ A or x ∈ B}. This operation on the elements of set A and B can be represented using a Venn diagram with two circles. The total region of both the circles combined denotes the union of sets A and B.
Intersection of Set Venn Diagram
The intersection of sets, A and B is given by: A ∩ B = {x : x ∈ A and x ∈ B}. This operation on set A and B can be represented using a Venn diagram with two intersecting circles. The region common to both the circles denotes the intersection of set A and Set B.
Complement of Set Venn Diagram
The complement of any set A can be given as A'. This represents elements that are not present in set A and can be represented using a Venn diagram with a circle. The region covered in the universal set, excluding the region covered by set A, gives the complement of A.
Difference of Set Venn Diagram
The difference of sets can be given as, A  B. It is also referred to as a ‘relative complement’. This operation on sets can be represented using a Venn diagram with two circles. The region covered by set A, excluding the region that is common to set B, gives the difference of sets A and B.
We can observe the aboveexplained operations on sets using the figures given below,
Venn Diagram for Three Sets
Three sets Venn diagram is made up of three overlapping circles and these three circles show how the elements of the three sets are related. When a Venn diagram is made of three sets, it is also called a 3circle Venn diagram. In a Venn diagram, when all these three circles overlap, the overlapping parts contain elements that are either common to any two circles or they are common to all the three circles. Let us consider the below given example:
Here are some important observations from the above image:
 Elements in P and Q = elements in P and Q only plus elements in P, Q, and R.
 Elements in Q and R = elements in Q and R only plus elements in P, Q, and R.
 Elements in P and R = elements in P and R only plus elements in P, Q, and R.
How to Draw a Venn Diagram?
Venn diagrams can be drawn with unlimited circles. Since more than three becomes very complicated, we will usually consider only two or three circles in a Venn diagram. Here are the 4 easy steps to draw a Venn diagram:
 Step 1: Categorize all the items into sets.
 Step 2: Draw a rectangle and label it as per the correlation between the sets.
 Step 3: Draw the circles according to the number of categories you have.
 Step 4: Place all the items in the relevant circles.
Example: Let us draw a Venn diagram to show categories of outdoor and indoor for the following pets: Parrots, Hamsters, Cats, Rabbits, Fish, Goats, Tortoises, Horses.
 Step 1: Categorize all the items into sets (Here, its pets): Indoor pets: Cats, Hamsters, and, Parrots. Outdoor pets: Horses, Tortoises, and Goats. Both categories (outdoor and indoor): Rabbits and Fish.
 Step 2: Draw a rectangle and label it as per the correlation between the two sets. Here, let's label the rectangle as Pets.
 Step 3: Draw the circles according to the number of categories you have. There are two categories in the sample question: outdoor pets and indoor pets. So, let us draw two circles and make sure the circles overlap.
 Step 4: Place all the pets in the relevant circles. If there are certain pets that fit both the categories, then place them at the intersection of sets , where the circles overlap. Rabbits and fish can be kept as indoor and outdoor pets, and hence they are placed at the intersection of both circles.
 Step 5: If there is a pet that doesn't fit either the indoor or outdoor sets, then place it within the rectangle but outside the circles.
Venn Diagram Formula
For any two given sets A and B, the Venn diagram formula is used to find one of the following: the number of elements of A, B, A U B, or A ⋂ B when the other 3 are given. The formula says:
n(A U B) = n(A) + n(B) – n (A ⋂ B)
Here, n(A) and n(B) represent the number of elements in A and B respectively. n(A U B) and n(A ⋂ B) represent the number of elements in A U B and A ⋂ B respectively. This formula is further extended to 3 sets as well and it says:
 n (A U B U C) = n(A) + n(B) + n(C)  n(A ⋂ B)  n(B ⋂ C)  n(C ⋂ A) + n(A ⋂ B ⋂ C)
Here is an example of Venn diagram formula.
Example: In a cricket school, 12 players like bowling, 15 like batting, and 5 like both. Then how many players like either bowling or batting.
Let A and B be the sets of players who like bowling and batting respectively. Then
n(A ⋂ B) = 5
We have to find n(A U B). Using the Venn diagram formula,
n(A U B) = 12 + 15  5 = 22.
Applications of Venn Diagram
There are several advantages to using Venn diagrams. Venn diagram is used to illustrate concepts and groups in many fields, including statistics, linguistics, logic, education, computer science, and business.
 We can visually organize information to see the relationship between sets of items, such as commonalities and differences, and to depict the relations for visual communication.
 We can compare two or more subjects and clearly see what they have in common versus what makes them different. This might be done for selecting an important product or service to buy.
 Mathematicians also use Venn diagrams in math to solve complex equations.
 We can use Venn diagrams to compare data sets and to find correlations .
 Venn diagrams can be used to reason through the logic behind statements or equations .
☛ Related Articles:
Check out the following pages related to Venn diagrams:
 Operations on Sets
 Roster Notation
 Set Builder Notation
 Probability
Important Notes on Venn Diagrams:
Here is a list of a few points that should be remembered while studying Venn diagrams:
 Every set is a subset of itself i.e., A ⊆ A.
 A universal set accommodates all the sets under consideration.
 If A ⊆ B and B ⊆ A, then A = B
 The complement of a complement is the given set itself.
Examples of Venn Diagram
Example 1: Let us take an example of a set with various types of fruits, A = {guava, orange, mango, custard apple, papaya, watermelon, cherry}. Represent these subsets using sets notation: a) Fruit with one seed b) Fruit with more than one seed
Solution: Among the various types of fruit, only mango and cherry have one seed.
Answer: a) Fruit with one seed = {mango, cherry} b) Fruit with more than one seed = {guava, orange, custard apple, papaya, watermelon}
Note: If we represent these two sets on a Venn diagram, the intersection portion is empty.
Example 2: Let us take an example of two sets A and B, where A = {3, 7, 9} and B = {4, 8}. These two sets are subsets of the universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9}. Find A ∪ B.
Solution: The Venn diagram for the above relations can be drawn as:
Answer: A ∪ B means, all the elements that belong to either set A or set B or both the sets = {3, 4, 7, 8, 9}
Example 3: Using Venn diagram, find X ∩ Y, given that X = {1, 3, 5}, Y = {2, 4, 6}.
Given: X = {1, 3, 5}, Y = {2, 4, 6}
The Venn diagram for the above example can be given as,
Answer: From the blue shaded portion of Venn diagram, we observe that, X ∩ Y = ∅ ( null set ).
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FAQs on Venn Diagrams
What is a venn diagram in math.
In math, a Venn diagram is used to visualize the logical relationship between sets and their elements and helps us solve examples based on these sets.
How do You Read a Venn Diagram?
These are steps to be followed while reading a Venn diagram:
 First, observe all the circles that are present in the entire diagram.
 Every element present in a circle is its own item or data set.
 The intersecting or the overlapping portions of the circles contain the items that are common to the different circles.
 The parts that do not overlap or intersect show the elements that are unique to the different circle.
What is the Importance of Venn Diagram?
Venn diagrams are used in different fields including business, statistics, linguistics, etc. Venn diagrams can be used to visually organize information to see the relationship between sets of items, such as commonalities and differences, and to depict the relations for visual communication.
What is the Middle of a Venn Diagram Called?
When two or more sets intersect, overlap in the middle of a Venn diagram, it is called the intersection of a Venn diagram. This intersection contains all the elements that are common to all the different sets that overlap.
How to Represent a Universal Set Using Venn Diagram?
A large rectangle is used to represent the universal set and it is usually denoted by the symbol E or sometimes U. All the other sets are represented by circles or closed figures within this larger rectangle that represents the universal set.
What are the Different Types of Venn Diagrams?
The different types of Venn diagrams are:
 Twoset Venn diagram: The simplest of the Venn diagrams, that is made up of two circles or ovals of different sets to show their overlapping properties.
 Threeset Venn diagram: These are also called the threecircle Venn diagram, as they are made using three circles.
 Fourset Venn diagram: These are made out of four overlapping circles or ovals.
 Fiveset Venn diagram: These comprise of five circles, ovals, or curves. In order to make a fiveset Venn diagram, you can also pair a threeset diagram with repeating curves or circles.
What are the Different Fields of Applications of Venn Diagrams?
There are different cases of applications of Venn diagrams: Set theory, logic, mathematics, businesses, teaching, computer science, and statistics.
Can a Venn Diagram Have 2 Non Intersecting Circles?
Yes, a Venn digram can have two non intersecting circles where there is no data that is common to the categories belonging to both circles.
What is the Formula of Venn Diagram?
The formula that is very helpful to find the unknown information about a Venn diagram is n(A U B) = n(A) + n(B) – n (A ⋂ B), where
 A and B are two sets.
 n(A U B) is the number of elements in A U B.
 n (A ⋂ B) is the number of elements in A ⋂ B.
Can a Venn Diagram Have 3 Circles?
Yes, a Venn diagram can have 3 circles , and it's called a threeset Venn diagram to show the overlapping properties of the three circles.
What is Union in the Venn Diagram?
A union is one of the basic symbols used in the Venn diagram to show the relationship between the sets. A union of two sets C and D can be shown as C ∪ D, and read as C union D. It means, the elements belong to either set C or set D or both the sets.
What is A ∩ B Venn Diagram?
A ∩ B (which means A intersection B) in the Venn diagram represents the portion that is common to both the circles related to A and B. A ∩ B can be a null set as well and in this case, the two circles will either be nonintersecting or can be represented with intersecting circles having no data in the intersection portion.
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How to Solve Problems Using Venn Diagrams
Venn diagrams are visual tools often used to organize and understand sets and the relationships between them. They're named after John Venn, a British philosopher, and logician who introduced them in the 1880s. Venn diagrams are frequently used in various fields, including mathematics, statistics, logic, computer science, etc. They're handy for solving problems involving sets and subsets, intersections, unions, and complements.
A Stepbystep Guide to Solving Problems Using Venn Diagrams
Here’s a stepbystep guide on how to solve problems using Venn diagrams:
Step 1: Understand the Problem
As with any problemsolving method, the first step is to understand the problem. What sets are involved? How are they related? What are you being asked to find?
Step 2: Draw the Diagram
Draw a rectangle to represent the universal set, which includes all possible elements. Each set within the universal set is represented by a circle. If there are two sets, draw two overlapping circles. If there are three sets, draw three overlapping circles, and so forth. Each section in the overlapping circles represents different intersections of the sets.
Step 3: Label the Diagram
Each circle (set) should be labeled appropriately. If you’re dealing with sets of different types of fruits, for example, one might be labeled “Apples” and another “Oranges”.
Step 4: Fill in the Values
Start filling in the values from the innermost part of the diagram (where all sets overlap) to the outer parts. This helps to avoid doublecounting elements that belong to more than one set. Information provided in the problem usually tells you how many elements are in each set or section.
Step 5: Solve the Problem
Now, you can use the diagram to answer the question. This might involve counting the number of elements in a particular set or section of the diagram, or it might involve noticing patterns or relationships between the sets.
Step 6: Check Your Answer
Make sure your answer makes sense in the context of the problem and that you’ve accounted for all elements in the diagram.
by: Effortless Math Team about 1 year ago (category: Articles )
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Venn Diagram Questions
Venn diagram questions with solutions are given here for students to practice various questions based on Venn diagrams. These questions are beneficial for both school examinations and competitive exams. Practising these questions will develop a skill to solve any problem on Venn diagrams quickly.
Venn diagrams were first introduced by John Venn to represent various propositions in a diagrammatic way. Venn diagrams are used for representing relationships between given sets. For example, natural numbers and whole numbers are subsets of integers represented by the Venn diagram:
Using Venn diagrams, we can easily understand whether given sets are subsets of each other or disjoint sets or have something in common.
 Intersection of Sets
 Union of Sets
 Complement of Set
 Set Operations
Following are some set operations and their meaning useful while solving problems on the Venn diagram:


A ⊂ B  Set A is a proper subset of B, or A is in B. 
A ⋃ B  Set of all those elements which to A to B 
A ∩ B  Set of all those elements which belong to both A B 
A or A’  Set of all those elements which are A 
A – B  Set of all those elements to A 
A ⊝ B  Symmetric difference: Set of all those elements which to A B, but . 
Some important formulae:
= ; is universal set = A ; is universal set = – A ) = A = φ 
Venn Diagram Questions with Solution
Let us practice some questions based on Venn diagrams.
Question 1: If A and B are two sets such that number of elements in A is 24, number of elements in B is 22 and number of elements in both A and B is 8, find:
(i) n(A ∪ B)
(ii) n(A – B)
(ii) n(B – A)
Given, n(A) = 24, n(B) = 22 and n(A ∩ B) = 8
The Venn diagram for the given information is:
(i) n(A ∪ B) = n(A) + n(B) – n(A ∩ B) = 24 + 22 – 8 = 38.
(ii) n(A – B) = n(A) – n(A ∩ B) = 24 – 8 = 16.
(iii) n(B – A) = n(B) – n(A ∩ B) = 22 – 8 = 14.
Question 2: According to the survey made among 200 students, 140 students like cold drinks, 120 students like milkshakes and 80 like both. How many students like atleast one of the drinks?
Number of students like cold drinks = n(A) = 140
Number of students like milkshake = n(B) = 120
Number of students like both = n(A ∩ B) = 80
Number of students like atleast one of the drinks = n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
= 140 + 120 – 80
Question 3: In a group of 500 people, 350 people can speak English, and 400 people can speak Hindi. Find how many people can speak both languages?
Let H be the set of people who can speak Hindi and E be the set of people who can speak English. Then,
n(H ∪ E) = 500
We have to find n(H ∩ E).
Now, n(H ∪ E) = n(H) + n(E) – n(H ∩ E)
⇒ 500 = 400 + 350 – n(H ∩ E)
⇒ n(H ∩ E) = 750 – 500 = 250.
∴ 250 people can speak both languages.
Questions 4: The following Venn diagram shows games played by the number of students in a class:
How many students like only cricket and only football?
As per the given Venn diagram,
Number of students only like cricket = 7
Number of students only like football = 14
∴ Number of students like only cricket and only football = 7 + 14 = 21.
Question 5: In a class of 40 students, 20 have chosen Mathematics, 15 have chosen mathematics but not biology. If every student has chosen either mathematics or biology or both, find the number of students who chose both mathematics and biology and the number of students chose biology but not mathematics.
Let, M ≡ Set of students who chose mathematics
B ≡ Set of students who chose biology
n(M ∪ B) = 40
n(B) = n(M ∪ B) – n(M)
⇒ n(B) = 40 – 20 = 20
n(M – B) = 15
n(M) = n(M – B) + n(M ∩ B)
⇒ 20 = 15 + n(M ∩ B)
⇒ n(M ∩ B) = 20 – 15 = 5
n(B – M) = n(B) – n(M ∩ B)
⇒ n(B – M) = 20 – 5 = 15
Question 6: Represent The following as Venn diagram:
(i) A’ ∩ (B ∪ C)
(ii) A’ ∩ (C – B)
Question 7: In a survey among 140 students, 60 likes to play videogames, 70 likes to play indoor games, 75 likes to play outdoor games, 30 play indoor and outdoor games, 18 like to play video games and outdoor games, 42 play video games and indoor games and 8 likes to play all types of games. Use the Venn diagram to find
(i) students who play only outdoor games
(ii) students who play video games and indoor games, but not outdoor games.
Let V ≡ Play video games
I ≡ Play indoor games
O ≡ Play outdoor games
n(V) = 60, n(I) = 70, n(O) = 75
n(I ∩ O) = 30, n(V ∩ O) = 18, n(V ∩ I) = 42
n(V ∩ I ∩ O) = 8
Hence, by Venn diagram
Number of students only like to play only outdoor games = 35
Number of students like to play video games and indoor games but not outdoor games = 34
Note : Always begin to fill the Venn diagram from the innermost part.
Question 8: Using the Venn diagrams, verify (P ∩ Q) ∪ R = (P ∪ R) ∩ (Q ∪ R).
The shaded portion represents (P ∩ Q) ∪ R in the Venn diagram.
Comparing both the shaded portion in both the Venn diagram, we get (P ∩ Q) ∪ R = (P ∪ R) ∩ (Q ∪ R).
Question 9: Prove using the Venn diagram: (B – A) ∪ (A ∩ B) = B.
From the Venn diagram, it is clear that (B – A) ∪ (A ∩ B) = B
Question 10: In a survey, it is found that 21 people read English newspaper, 26 people read Hindi newspaper, and 29 people read regional language newspaper. If 14 people read both English and Hindi newspapers; 15 people read both Hindi and regional language newspapers; 12 people read both English and regional language newspaper and 8 read all types of newspapers, find:
(i) How many people were surveyed?
(ii) How many people read only regional language newspapers?
Let A ≡ People who read English newspapers.
B ≡ People who read Hindi newspapers.
C ≡ People who read Hindi newspapers.
n(A) = 21, n(B) = 26, n(C) = 29
n(A ∩ B) = 14, n(B ∩ C) = 15, n(A ∩ C) = 12
n(A ∩ B ∩ C) = 8
(i) Number of people surveyed = n(A ∪ B ∪ C) = n(A) + n(B) + n(C) – n(A ∩ B) – n(B ∩ C) – n(A ∩ C) + n(A ∩ B ∩ C) = 21 + 26 + 29 – 14 – 15 – 12 + 8 = 43
(ii) By the Venn diagram, number of people who only read regional language newspapers = 10.
Video Lesson on Introduction to Sets




Practice Questions on Venn Diagrams
1. Verify using the Venn diagram:
(i) A – B = A ∩ B C
(ii) (A ∩ B) C = A C ∪ B C
2. For given two sets P and Q, n(P – Q) = 24, n(Q – P) = 19 and n(P ∩ Q) = 11, find:
(iii) n (P ∪ Q)
3. In a group of 65 people, 40 like tea and 10 like both tea and coffee. Find
(i) how many like coffee only and not tea?
(ii) how many like coffee?
4. In a sports tournament, 38 medals were awarded for 500 m sprint, 15 medals were awarded for Javelin throw, and 20 medals were awarded for a long jump. If these medals were awarded to 58 participants and among them only three medals in all three sports, how many received exactly two medals?
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Section SV.3 – Venn Diagrams
To visualize the interaction of sets, John Venn in 1880 thought to use overlapping circles, building on a similar idea used by Leonhard Euler in the 18 th century. These illustrations are now called Venn Diagrams .
A Venn diagram represents each set by a circle, usually drawn inside of a rectangle. The rectangle represents the universal set. Overlapping areas of circles indicate elements common to both sets. Note: There is no meaning to the size of the circle. Basic Venn diagrams can illustrate the interaction of two or three sets. 
Create Venn diagrams to illustrate ⋃ , ⋂ , and ⋂  
⊂ means that A is a proper subset of set B. So all elements in set A are also elements in Set B.
 ⋃ contains all elements in set. The elements can be in set A or set B or both.

⋂ contains only those elements in both sets – in the overlap of the circles. The elements are in set A and set B.
 will contain all elements in the set A. ⋂ will contain the elements in set that are not in set .

Draw a Venn Diagram to represent that set G is a proper subset of set P. 
Draw a Venn Diagram to represent the sets U = {a, b, c, d, e, f, g}, A = {a, b, d, e}, and B = {d, e, g}. 
We’ll start by identifying all the elements in the set ⋂ This means all the elements in set A and set B.
Now, write the remaining elements of A inside the region A but outside the intersection of A and B. Do the same for the remaining elements of B.
Now, complete the Venn Diagram by writing any elements from the universal set that were not in A or B.

Use a Venn diagram to illustrate [latex]\overline{(H~\cap~F)}~\cap~W[/latex]. 
We’ll start by identifying everything in the set ⋂
Now, [latex]\overline{(H~\cap~F)}~\cap~W[/latex] will contain everything in the set identified above that is also in set .

Create an expression to represent the outlined part of the Venn diagram shown.

The elements in the outlined set in sets and , but are not in set . So we could represent this set as ⋂ ⋂ 
Create an expression to represent the outlined portion of the Venn diagram shown below.

Use the Venn diagram to write each of the sets below in roster form.

• A = All the elements in the circle A: {a, d, e, f, g, m} • A ∪ B = All the elements in the circles A and B: {a, d, e, f, g, m, h, c} • [latex]\overline{A~\cup~B~\cup~C}[/latex] = All the elements that are not in A, B or C: {v, j} • A ∩ B ∩ C = All the elements that are in all three sets. The center region where all three sets overlap is empty: { } or ∅ • (A ∪ B) ∩ C̅ = We need the elements in A or B combined and not in C: {a, d, e, f, g, h} 
Section SV.3 – Answers to You Try Problems
G is a proper subset of P, so G must be completely contained within P.
( A ∪ B ) ∩ C̅
College Mathematics  MAT14X  3rd Edition Copyright © by Adam Avilez; Shelley Ceinaturaga; and Terri D. Levine is licensed under a Creative Commons AttributionNonCommercial 4.0 International License , except where otherwise noted.
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Set Theory: Venn Diagrams And Subsets
Related Pages Union Of Sets Intersection Of Two Sets Intersection Of Three Sets More Lessons On Sets More Lessons for GCSE Maths Math Worksheets
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What Is A Venn Diagram?
A Venn Diagram is a pictorial representation of the relationships between sets.
We can represent sets using Venn diagrams . In a Venn diagram, the sets are represented by shapes; usually circles or ovals. The elements of a set are labeled within the circle.
The following diagrams show the set operations and Venn Diagrams for Complement of a Set, Disjoint Sets, Subsets, Intersection and Union of Sets. Scroll down the page for more examples and solutions.
The set of all elements being considered is called the Universal Set (U) and is represented by a rectangle.
 The complement of A, A’ , is the set of elements in U but not in A. A’ ={ x  x ∈ U and x ∉ A}
 Sets A and B are disjoint sets if they do not share any common elements.
 B is a proper subset of A. This means B is a subset of A, but B ≠ A.
 The intersection of A and B is the set of elements in both set A and set B. A ∩ B = { x  x ∈ A and x ∈ B}
 The union of A and B is the set of elements in set A or set B. A ∪ B = { x  x ∈ A or x ∈ B}
Set Operations And Venn Diagrams
Example: 1. Create a Venn Diagram to show the relationship among the sets. U is the set of whole numbers from 1 to 15. A is the set of multiples of 3. B is the set of primes. C is the set of odd numbers.
2. Given the following Venn Diagram determine each of the following set. a) A ∩ B b) A ∪ B c) (A ∪ B)’ d) A’ ∩ B e) A ∪ B'
Venn Diagram Examples
Example: Given the set P is the set of even numbers between 15 and 25. Draw and label a Venn diagram to represent the set P and indicate all the elements of set P in the Venn diagram.
Solution: List out the elements of P . P = {16, 18, 20, 22, 24} ← ‘between’ does not include 15 and 25 Draw a circle or oval. Label it P . Put the elements in P .
Example: Draw and label a Venn diagram to represent the set R = {Monday, Tuesday, Wednesday}.
Solution: Draw a circle or oval. Label it R . Put the elements in R .
Example: Given the set Q = { x : 2 x – 3 < 11, x is a positive integer }. Draw and label a Venn diagram to represent the set Q .
Solution: Since an equation is given, we need to first solve for x . 2 x – 3 < 11 ⇒ 2 x < 14 ⇒ x < 7
So, Q = {1, 2, 3, 4, 5, 6} Draw a circle or oval. Label it Q . Put the elements in Q .
Venn Diagram Videos
What’s a Venn Diagram, and What Does Intersection and Union Mean?
Venn Diagram and Subsets
Learn about Venn diagrams and subsets.
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Venn diagram
A Venn diagram is a visual way of representing the mathematical relationship between sets .
 1 Two Set Example
 2 Three Set Example
 3 Using Venn Diagrams
 4 External links
Two Set Example
All of this information can be summarized in the following table:
Region (by color)  Description  Notation 

Red  elements in  
Blue  elements in  
Black  elements in both  
Gray  elements in neither  
or  
or  
or 
Three Set Example
The following table describes the various regions in the diagram:
Region (by color)  Description  Notation 

Blue  elements in  
Yellow  elements in  
Red  elements in  
Green  elements in both  
Orange  elements in both  
Purple  elements in  
Black  elements in  
Gray  elements in neither  
or  
or  
or 
Using Venn Diagrams
Venn diagrams are very useful for organizing data. In particular, the Principle of InclusionExclusion can be explained for small cases nicely using them.
External links
 A Survey of Venn Diagrams
 Combinatorics
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Venn diagram word problems
The Venn diagram word problems in this lesson will show you how to use Venn diagrams with 2 circles to solve problems involving counting.
Venn diagram word problems with two circles
Word problem #1
A survey was conducted in a neighborhood with 128 families. The survey revealed the following information.
 106 of the families have a credit card
 73 of the families are trying to pay off a car loan
 61 of the families have both a credit card and a car loan
Answer the following questions:
1. How many families have only a credit card?
2. How many families have only a car loan?
3. How many families have neither a credit card nor a car loan?
4. How many families do not have a credit card?
5. How many families do not have a car loan?
6. How many families have a credit card or a car loan?
 Let C be families with a credit card
 Let L be families with a car loan
 Let S be the total number of families
The Venn diagram above can be used to answer all these questions.
Tips on how to create the Venn diagram. Always put first , in the middle or in the intersection, the value that is in both sets. For example, since 61 families have both a credit card and a car loan, put 61 in the intersection before you do anything else. In C only, put 45 since 106  61 = 45
In L only, put 12 since 73  61 = 12
Outside C and L, put 10 since 128  61  45  12 = 10
The expression, " only a credit card" means that it is only in C. Any number in L cannot be included. 1. The number of families with only a credit card is 45. Do not add 61 to 45 since 61 is in L.
2. The number of families with only a car loan is 12.
3. The number of families with neither a credit card nor a car loan is 10. 10 is not in C nor in L.
4. The number families without a credit card is found by adding everything that is not in C. 12 + 10 = 22
5. The number families without a car loan is found by adding everything that is not in L. 45 + 10 = 55
6. The number of families with a credit card or a car loan is found by adding anything in C only, in L only and in the intersection of C and L?
45 + 61 + 12 = 118
Word problem #2
A survey conducted in a school with 150 students revealed the following information:
 78 students are enrolled in swimming class
 85 students are enrolled in basketball class
 25 are enrolled in both swimming and basketball class
1. How many students are enrolled only in swimming class?
2. How many students are enrolled only in basketball class?
3. How many students are neither enrolled in swimming class nor basketball class?
4. How many students are not enrolled in swimming class?
5. How many students are not enrolled in basketball class?
6. How many students are enrolled in swimming class or basketball class?
 Let S be students enrolled in swimming class
 Let B be students enrolled in basketball class
 Let E be the total number of students
Using the same technique as in problem #1 , we have the following Venn diagram
1. The number of students enrolled only in swimming class is 53 2. The number of students enrolled only in basketball class is 60
3. The number of students who are neither enrolled in swimming class nor basketball class is 12
4. Students not enrolled in swimming class are enrolled in basketball class only or are enrolled in neither of these two activities. In other words, everything that is not in S.
60 + 12 = 72
5. Students not enrolled in basketball class are enrolled in swimming class only or are enrolled in neither of these two activities. In other words, everything that is not in B.
53 + 12 = 65
6. The number of students enrolled in swimming class or basketball class is found by adding anything in S only, in B only and in the intersection of S and B?
53 + 25 + 60 = 138
A tricky Venn diagram word problem with two circles
Word problem #3
In a survey of 100 people, 28 people smoke, 65 people drink, and 30 people do neither. How many people do both?
 Let K be the number of people who smoke
 Let D be the number of people who drink
 Let E be the total number of people
 Let x be the number of people who smoke and drink
If we make a Venn diagram, here is what we have so far.
We end up with the following equation to solve for x.
(65  x) + x + (28  x) + 30 = 100
65  x + x + 28  x + 30  30 = 100  30
65  x + x + 28  x = 70
65 + 0 + 28  x = 70
93  x = 70
Since 93  23 = 70, x = 23
The number of people who do both is 23.
3circle Venn diagram
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Venn Diagrams: Exercises
Intro Set Not'n Sets Exercises Diag. Exercises
Venn diagram word problems generally give you two or three classifications and a bunch of numbers. You then have to use the given information to populate the diagram and figure out the remaining information. For instance:
Out of forty students, 14 are taking English Composition and 29 are taking Chemistry.
 If five students are in both classes, how many students are in neither class?
 How many are in either class?
 What is the probability that a randomlychosen student from this group is taking only the Chemistry class?
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There are two classifications in this universe: English students and Chemistry students.
First I'll draw my universe for the forty students, with two overlapping circles labelled with the total in each:
(Well, okay; they're ovals, but they're always called "circles".)
Five students are taking both classes, so I'll put " 5 " in the overlap:
I've now accounted for five of the 14 English students, leaving nine students taking English but not Chemistry, so I'll put " 9 " in the "English only" part of the "English" circle:
I've also accounted for five of the 29 Chemistry students, leaving 24 students taking Chemistry but not English, so I'll put " 24 " in the "Chemistry only" part of the "Chemistry" circle:
This tells me that a total of 9 + 5 + 24 = 38 students are in either English or Chemistry (or both). This gives me the answer to part (b) of this exercise. This also leaves two students unaccounted for, so they must be the ones taking neither class, which is the answer to part (a) of this exercise. I'll put " 2 " inside the box, but outside the two circles:
The last part of this exercise asks me for the probability that a agiven student is taking Chemistry but not English. Out of the forty students, 24 are taking Chemistry but not English, which gives me a probability of:
24/40 = 0.6 = 60%
 Two students are taking neither class.
 There are 38 students in at least one of the classes.
 There is a 60% probability that a randomlychosen student in this group is taking Chemistry but not English.
Years ago, I discovered that my (now departed) cat had a taste for the adorable little geckoes that lived in the bushes and vines in my yard, back when I lived in Arizona. In one month, suppose he deposited the following on my carpet:
 six gray geckoes,
 twelve geckoes that had dropped their tails in an effort to escape capture, and
 fifteen geckoes that he'd chewed on a little
In addition:
 only one of the geckoes was gray, chewedon, and tailless;
 two were gray and tailless but not chewedon;
 two were gray and chewedon but not tailless.
If there were a total of 24 geckoes left on my carpet that month, and all of the geckoes were at least one of "gray", "tailless", and "chewedon", how many were tailless and chewedon, but not gray?
If I work through this stepbystep, using what I've been given, I can figure out what I need in order to answer the question. This is a problem that takes some time and a few steps to solve.
They've given me that each of the geckoes had at least one of the characteristics, so each is a member of at least one of the circles. This means that there will be nothing outside of the circles; the circles will account for everything in this particular universe.
There was one gecko that was gray, tailless, and chewed on, so I'll draw my Venn diagram with three overlapping circles, and I'll put " 1 " in the center overlap:
Two of the geckoes were gray and tailless but not chewedon, so " 2 " goes in the rest of the overlap between "gray" and "tailless".
Two of them were gray and chewedon but not tailless, so " 2 " goes in the rest of the overlap between "gray" and "chewedon".
Since a total of six were gray, and since 2 + 1 + 2 = 5 of these geckoes have already been accounted for, this tells me that there was only one left that was only gray.
This leaves me needing to know how many were tailless and chewedon but not gray, which is what the problem asks for. But, because I don't know how many were only chewed on or only tailless, I cannot yet figure out the answer value for the remaining overlap section.
I need to work with a value that I don't yet know, so I need a variable. I'll let " x " stand for this unknown number of tailless, chewedon geckoes.
I do know the total number of chewed geckoes ( 15 ) and the total number of tailless geckoes ( 12 ). After subtracting, this gives me expressions for the remaining portions of the diagram:
only chewed on:
15 − 2 − 1 − x = 12 − x
only tailless:
12 − 2 − 1 − x = 9 − x
There were a total of 24 geckoes for the month, so adding up all the sections of the diagram's circles gives me: (everything from the "gray" circle) plus (the unknown from the remaining overlap) plus (the onlychewedon) plus (the onlytailless), or:
(1 + 2 + 1 + 2) + ( x )
+ (12 − x ) + (9 − x )
= 27 − x = 24
Solving , I get x = 3 . So:
Three geckoes were tailless and chewed on but not gray.
(No geckoes or cats were injured during the production of the above word problem.)
For more wordproblem examples to work on, complete with worked solutions, try this page provided by Joe Kahlig of Texas A&M University. There is also a software package (DOSbased) available through the Math Archives which can give you lots of practice with the settheory aspect of Venn diagrams. The program is not hard to use, but you should definitely read the instructions before using.
URL: https://www.purplemath.com/modules/venndiag4.htm
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Venn Diagram Word Problems
Venn Diagram Word Problems can be very easy to make mistakes on when you are a beginner.
It is extremely important to:
Read the question carefully and note down all key information.
Know the standard parts of a Venn Diagram
Work in a step by step manner
Check at the end that all the numbers add up coorectly.
Let’s start with an easy example of a two circle diagram problem.
Venn Diagrams – Word Problem One
“A class of 28 students were surveyed and asked if they ever had dogs or cats for pets at home. 8 students said they had only ever had a dog. 6 students said they had only ever had a cat. 10 students said they had a dog and a cat. 4 students said they had never had a dog or a cat.”
Note that the word “only” is extremely important in Venn Diagram word problems.
Because the word “only” is in our problem text, it makes it an easy word problem. Since this question is about dogs and cats, it will require a two circle Venn Diagram.
Here is the type of diagram we will need.
Our problem is an easy one where we have been given all of the numbers for the items required on the diagram.
We do not need to work out any missing values.
All we need to do is place the numbers from the word problem onto the standard Venn Diagram and we are done.
Venn Diagrams – Word Problem Two
The answer for this question will actually be the same as the Cats and Dogs question in Example 1.
However this time we are given less information, and so we will have work out the missing information.
Here is Problem 2: “A class of 28 students were surveyed and asked if they ever had dogs or cats for pets at home. 18 students said they had a dog. 16 students said they had a cat. 4 students said they had never had a dog or a cat.”
The above question does not contain the word “only” anywhere in it, and this is an indication that we will have to do some working out.
The question states that: “18 students said they had a dog” without the word “only” in there.
This means that the total of the Dogs circle is 18.
The 18 total students for Dogs includes people that have both a cat and a dog, as well as people who only have a dog.
Some people, who do not read this question carefully, will simply take the above figures and put them straight into a Venn Diagram like this.
Always check at the end that the numbers add up to the “E” Grand total.
16 + 18 + 4 = 38 which is much bigger than the “E” total of 28.
From the given information we have been able to work out that the circles total is 24. (Eg. Everything Total – No Cats and No Dogs = 28 – 4 = 24. This is vital information we now use to work on the rest of the problem.
Let’s first work out the “Only Cats” value.
Next we work out the “Only Dogs” number of people.
All we have left to work out is the number of Cats and Dogs for the center of the diagram.
We can do this any of three possible ways: Cats and Dogs = Total Cats – Only Cats or
Cats and Dogs = Total Dogs – Only Dogs
Cats and Dogs = E Total – Only Cats – Only Dogs – (No cats and No Dogs)
Any way that we work it out, the answer is 10.
So here is the final completed Venn Diagram Answer.
When putting answers into our Mathematics Workbook, we do not have to color in the diagram.
A final answer like the following is quite acceptable.
We can summarise the steps we used to work out this problem as follows.
Word Problem Two – Summary of Steps
– Work out What Information is given, and what needs to be calculated.
– Circles Total = E everything – (No Cats and No Dogs) – Cats Only = Circles Total – Total Dogs
– Dogs Only = Circles Total – Total Cats
– Cats and Dogs = Cats Total – Cats Only
– Finally, check that all the numbers in the diagram add up to equal the “E” everything total.
Word Problem Three – Subsets
“Fifty people were surveyed and only 20 people said that they regularly eat Healthy Foods like Fruit and Vegetables. Of these 20 healthy eaters, 12 said that they ate Vegetables every day. Draw a Venn Diagram to represent these results.”
This problem is quite different to our other two circle diagrams.
Cats and Dogs are very different to each other, and so we needed two separate circles.
However Healthy Foods and Vegetables are not different to each other because Vegetables are a type of Healthy Food.
We say that vegetables are a “Subset” of Healthy Foods.
This means that we do not separate the circles. We actually need to draw our circles inside each other like this.
The total adds up to 50, and the 12 people who include vegetables in their healthy foods are shown as being fully inside the Healthy Foods circle.
Word Problem Four – Disjoint Sets “Draw a Venn Diagram which divides the twelve months of the year into the following two groups: Months whose name begins with the letter “J” and Months whose name ends in “ber”. You will need a two circle Venn Diagram for your answer.” The first step is to list the twelve months of the year:
January – named after Janus, the god of doors and gates February – named after Februalia, when sacrifices were made for sins March – named after Mars, the god of war April – from aperire, Latin for “to open” (buds) May – named after Maia, the goddess of growth of plants June – named after junius, Latin for the goddess Juno July – named after Julius Caesar in 44 B.C. August – named after Augustus Caesar in 8 B.C. September – from septem, Latin for “seven” October – from octo, Latin for “eight” November – from novem, Latin for “nine” December – from decem, Latin for “ten”
Months starting with J = { January, June, July }
Months ending in “ber” = { September, October, November, December }
The two sets do not have any items in common, and so we will not overlap them. The remaining months will need to go outside of our two circles.
There should be all twelve months in the diagram when we are finished.
The completed Venn Diagram is shown below:
Venn Word Problems – Summary We have not included three circle diagrams, as they will be covered in a separate lesson.
Remember the working out steps for harder problems are:
Work out What Information is given, and what needs to be calculated.
Check to see if the two sets are “Subsets” or “Disjoint” sets.
If they are “Intersecting Sets” then some of the following formulas may be needed.
Circles Total = E everything – (Not in A and Not in B)
In A Only = Both Circles Total – Total in B
In A Only = The A Circle Total – Total in the intersection (A and B)
In B Only = Both Circles Total – Total in A
In B Only = The B Circle Total – Total in the intersection (A and B)
In the Intersection (A and B) = Total in B – In B Only
In the Intersection (A and B) = Total in A – In A Only
Finally, check that the numbers in the diagram all add up to equal the “E” everything total.
Venn Word Problems Videos
The following video shows a typical two circles word problem.
Here is a video that covers a two circles problem, where we need to find the number of items that are ( not in “A” and not in “B”)
Here is a Video which shows how to solve Venn Diagram Survey Problems.
Related Items
Introduction to Venn Diagrams Three Circle Venn Diagrams Real World Venn Diagrams
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Venn Diagrams
 Practice Questions
Venn Diagrams with our comprehensive guide, tailored for educators and students striving for clarity in mathematics and logic. This guide demystifies the concept of Venn Diagrams, showcasing their utility in representing logical relationships visually. Through practical examples and simplified explanations, we aim to enhance analytical thinking and problemsolving skills. Whether it’s for classroom instruction or selfstudy, this resource is your goto for mastering the intricacies of Venn Diagrams.
What are Venn Diagrams?
Venn Diagrams are powerful tools used to illustrate the mathematical or logical relationship between different sets. By drawing circles that overlap, they visually represent how items or groups share common traits or differ, making complex relationships easier to understand. This method is invaluable in various fields, including statistics, logic, and education, helping students and teachers alike to dissect and comprehend the interconnectedness of concepts.
What is the Best Example of a Venn Diagram?
A prime example of a Venn Diagram is comparing characteristics of mammals and reptiles. By placing unique traits in separate circles and shared traits in the overlapping area, students can visually discern the similarities and differences between the two classes. This not only aids in grasping biological classifications but also in developing critical thinking skills by analyzing how groups relate to each other.
Venn Diagram Formula
Delve into the Venn Diagram formula with this succinct guide, designed to make mathematical concepts accessible for educators and students. The formula, helps quantify the elements in union and intersection of two sets, A and B. This formula is a cornerstone for understanding set theory, providing a mathematical framework to solve problems involving elements from multiple sets. Through clear examples and applications, this description empowers users to apply the Venn Diagram formula effectively in various scenarios.
n(A U B) = n(A) + n(B) – n (A ⋂ B)
Venn diagram example.
Unlock the potential of Venn Diagrams with this insightful guide, crafted to aid educators and students in visualizing complex relationships between sets. Venn Diagrams serve as a versatile tool in mathematics, logic, and beyond, illustrating intersections, unions, and differences with clarity. By providing a visual representation of how sets overlap, these diagrams facilitate a deeper understanding of concepts, enhancing analytical and critical thinking skills.
Example 1: Fruits and Citrus Fruits
A Venn Diagram showcasing the relationship between fruits in general and citrus fruits specifically. The circle for fruits includes apples, bananas, while the citrus circle includes oranges, lemons, with grapefruits appearing in the overlap, indicating it’s both a fruit and a citrus.
Example 2: Students in Sports Clubs
Illustrates students participating in basketball, football, and those who play both. Basketball and football circles overlap on students who play both, demonstrating the diagram’s ability to depict shared membership in two sets.
Example 3: Vegetarians and Vegans
This Venn Diagram differentiates between vegetarians and vegans. Vegetarians are represented in one circle, including eggs and cheese, while vegans are in another, focusing on plantbased foods. The overlap highlights foods both groups consume, like vegetables and fruits.
Example 4: Fiction and Science Fiction Books in a Library
Depicts the categorization of fiction and science fiction books. The fiction circle includes romance and historical novels, science fiction includes space operas, with dystopian novels in the overlap, indicating their dual categorization.
Example 5: WaterSoluble and FatSoluble Vitamins
A Venn Diagram to classify watersoluble (C and B vitamins) and fatsoluble vitamins (A, D, E, K). The diagram clearly separates them, with no overlap, indicating distinct absorption pathways in the body.
These examples underscore the Venn Diagram’s utility in teaching and learning, offering a visual method to compare and contrast various concepts across disciplines.
Terms Related to Venn Diagram
Dive into the world of Venn Diagrams with this essential terminology guide. Perfect for educators and students, it covers key terms necessary for understanding and applying Venn Diagrams in logical, mathematical, and statistical contexts. From “sets” to “intersections,” this guide enriches your vocabulary, enabling clearer communication and comprehension of complex relationships between groups.
 Explanation: In a Venn Diagram, a set is represented by a circle, encompassing its elements to visualize groupings in a problem or scenario.
 Explanation: Indicates when all elements of one set (subset) are also elements of another, shown by one circle entirely within another in a Venn Diagram.
 Explanation: The overlapping area of circles in a Venn Diagram, representing elements shared by sets A and B.
 Explanation: In Venn Diagrams, the union is depicted by the total area covered by both sets’ circles, including the intersection.
 Explanation: Represented in Venn Diagrams by the area outside a set’s circle but within the universal set boundary, indicating elements excluded from the set.
Venn Diagram Symbols
Master Venn Diagram symbols with this concise guide, tailored for educational use. These symbols form the visual language of Venn Diagrams, facilitating the representation of mathematical and logical relationships between sets. Understanding these symbols is crucial for teachers and students to effectively analyze and convey information through Venn Diagrams.
 Explanation: Each circle in a Venn Diagram encapsulates the elements of a set, serving as a visual boundary for group identification.
 Explanation: The area where circles intersect illustrates the common elements between sets, crucial for understanding their relationships.
 Explanation: Surrounding the circles, the rectangle defines the universe of discourse, including all elements considered in the scenario.
 Explanation: Dots placed inside circles signify individual members of sets, making it easy to count and identify specific elements.
 Explanation: Shaded areas outside a set but within the universal set boundary show the complement, or elements not included in the set under consideration.
Venn Diagram for Sets Operations
Explore set operations through Venn Diagrams with this insightful guide. Ideal for educational purposes, it simplifies complex set theories, making them accessible for teachers and students. Venn Diagrams visually represent set operations like unions, intersections, and complements, providing a clear method to understand and teach mathematical relationships between sets.
 Explanation: The Venn Diagram for a union showcases the total area covered by sets A and B, emphasizing inclusivity of elements.
 Explanation: Highlighted by the overlap between sets, the intersection focuses on elements shared by A and B.
 Explanation: Illustrated by shading outside A but within the universal set, this operation reveals elements excluded from set A.
 Explanation: This operation is visualized by shading parts of A’s circle that do not overlap with B, showing elements unique to A.
 Explanation: The Venn Diagram highlights the nonoverlapping parts of A and B, focusing on elements exclusive to each set, excluding their intersection.
Difference of Set Venn Diagram
Discover the essence of set differences with Venn Diagrams in this concise guide. Ideal for teachers and students, it illustrates how to visually represent the difference between two sets, A and B, by highlighting the elements that belong exclusively to set A. This concept is pivotal in understanding complex relationships in mathematics, fostering a deeper comprehension of how sets interact with each other.
 The difference shows prime numbers that are not even, like 3 and 5, visually excluding even primes like 2.
 Illustrates fruits that are not citrus, such as apples and bananas, by highlighting them outside the citrus section.
 Demonstrates that nonnovel books (e.g., dictionaries) belong to set A but not to set B.
 Showcases animals that are not mammals, like birds and fish, separated visually.
 Highlights vehicles that are not electric, indicating the broader category excludes the subset of electric vehicles.
Venn Diagram of Three Sets
Explore the dynamics of threeset Venn Diagrams with our guide, perfect for educators looking to depict the complex relationships between three distinct sets. This visualization tool sheds light on how sets intersect, combine, and differ, offering a comprehensive understanding of shared and unique elements. It’s an invaluable resource for enhancing analytical thinking and problemsolving skills in mathematics and logic.
 Reveals animals that are mammals, aquatic, both, or neither, providing insight into biological classifications.
 Depicts students who are exclusively in one subject or intersect in two or all, highlighting academic preferences.
 Shows how books can be categorized into fiction, mystery, both, and those that are bestsellers, revealing market trends.
 Illustrates the overlap between vegetables, foods that are green, and those high in fiber, offering dietary insights.
 Clarifies countries that are in Europe, part of the EU, the Eurozone, or a combination, enhancing geographical understanding.
How to Draw a Venn Diagram?
Master the art of drawing Venn Diagrams with our straightforward guide. Tailored for both teachers and students, this resource simplifies the process into manageable steps, from conceptualizing set relationships to visualizing intersections and differences. Whether for mathematical equations, logical reasoning, or categorizing information, learning to draw Venn Diagrams is an essential skill for effective problemsolving and communication.
 Begin by defining the sets you want to compare or contrast to understand their relationships.
 Draw circles to represent each set, ensuring they overlap for common elements.
 Clearly label each circle with the corresponding set name for easy identification.
 Place elements in the respective areas: unique in nonoverlapping and common in overlapping sections.
 Use the completed diagram to analyze and explain the relationships between the sets, such as intersections and differences.
These guides and examples are crafted to enhance the understanding and application of Venn Diagrams in educational settings, fostering a deeper comprehension of set theory and logical analysis among students.
Applications of Venn Diagram
Venn Diagrams serve as versatile tools in education, data analysis, and problemsolving. By visually mapping out relationships between sets, they facilitate understanding of complex concepts through overlap and distinction. Ideal for both classroom learning and professional data presentation, Venn Diagrams enhance comprehension in subjects ranging from mathematics to social sciences, making them indispensable in analytical reasoning and decisionmaking processes.
 Explanation: Students can visualize numbers that are exclusively prime or composite and those that share attributes with a Venn Diagram, enhancing number theory comprehension.
 Explanation: By using Venn Diagrams, teachers can help students identify common themes and unique elements in different works, promoting deeper literary analysis.
 Explanation: Venn Diagrams simplify the comparison of taxonomic groups, helping students grasp similarities and differences in traits among species.
 Explanation: Businesses utilize Venn Diagrams to understand shared characteristics among different customer segments, optimizing marketing strategies.
 Explanation: In conflict resolution, Venn Diagrams can highlight common ground and differing points, aiding in finding mutually acceptable solutions.
Venn Diagram Purpose and Benefits
Venn Diagrams are pivotal in simplifying the visualization of complex relationships, offering clarity in educational, analytical, and strategic contexts. They enable users to compare and contrast sets, highlighting similarities, differences, and intersections with ease. This visualization aids in fostering critical thinking, enhancing memory retention, and supporting effective communication, making Venn Diagrams a powerful tool for learners and professionals alike.
 Explanation: The visual representation in Venn Diagrams aids in memory retention by simplifying abstract concepts into tangible comparisons.
 Explanation: Venn Diagrams challenge students to think critically about how sets relate, enhancing logical reasoning skills.
 Explanation: In group settings, Venn Diagrams serve as focal points for discussion, promoting collaborative learning and idea exchange.
 Explanation: Venn Diagrams make it easier to present and interpret complex data, improving audience understanding in presentations.
 Explanation: By visually comparing options, Venn Diagrams help in weighing the pros and cons, facilitating more informed decisionmaking.
Venn Diagram Use Cases
Venn Diagrams are employed across various fields to visually organize information, facilitating understanding and analysis of relationships between sets. From educational settings to business strategy and scientific research, they provide a clear and intuitive method to display intersections, differences, and similarities, enhancing decisionmaking, learning, and data interpretation.
 Explanation: Teachers use Venn Diagrams to compare and contrast grammatical elements, helping students understand language rules.
 Explanation: Companies employ Venn Diagrams to compare their products with competitors’, identifying unique selling points and areas for improvement.
 Explanation: Researchers use Venn Diagrams to visualize similarities and differences in symptoms among diseases, aiding in differential diagnosis.
 Explanation: Venn Diagrams help students understand the shared and unique characteristics of different ecosystems, promoting environmental awareness.
 Explanation: Developers use Venn Diagrams to compare features across different software versions or competitors, guiding development priorities.
Intersection of Two Sets in Venn Diagram
Explore the concept of the intersection of two sets in Venn Diagrams, a crucial element for teachers and students in understanding shared characteristics between groups. This visualization technique marks the common elements of sets within the overlapping regions of circles, simplifying complex relationships and enhancing analytical reasoning. Perfect for classroom discussions, it facilitates a deeper comprehension of how sets interact in mathematics and logic.
 The intersection shows students involved in both sports, highlighting shared participants in the overlapping area.
 This intersection identifies individuals who enjoy both beverages, represented by the shared space between two circles.
 The overlapping section illustrates animals sharing traits of mammals and carnivores, aiding in biological classification.
 In the intersection, books categorized as fiction and mystery are shown, demonstrating how genres can overlap.
 This example uses the intersection to show countries that are part of Europe and the Schengen Agreement, facilitating a geographic and political understanding.
Union of Two Sets in Venn Diagram
The union of two sets in Venn Diagrams represents the combination of all elements from both sets, including the shared and unique elements. This concept is vital for students and teachers, offering a visual method to comprehend the totality of distinct and overlapping characteristics within groups. By enhancing visualization skills, it aids in grasping the breadth of set relationships, making it a fundamental tool in mathematical education.
 The union includes all books from both genres, emphasizing the extensive range of literature available.
 This union displays students taking either or both subjects, showcasing the diverse interests within the school population.
 The combined set includes all countries from both continents, highlighting the vast geographical coverage.
 By uniting the two sets, it shows individuals speaking either or both languages, reflecting linguistic diversity.
 The union encompasses all pets falling into either category, illustrating the variety of household animals.
Complement of Union of Sets in Venn Diagram
The complement of the union of sets in Venn Diagrams refers to elements not included in the union of specified sets, offering a unique perspective on set relationships. This concept is essential for educators teaching logical complementation, as it visually separates the universal set from the combined sets. By identifying elements outside the union, students gain insight into exclusion within set theory, enriching their understanding of mathematical and logical boundaries.
 The complement shows students who participate in neither activity, highlighting diverse interests outside these areas.
 This example identifies foods categorized outside of fruits and vegetables, emphasizing the variety in dietary choices.
 The complement includes books outside these genres, showcasing the wide range of literature beyond specific categories.
 By focusing on the complement, it reveals countries situated outside these continents, expanding geographic knowledge.
 This complement helps in understanding the diversity of animal kingdoms beyond avian and aquatic life forms.
Complement of Intersection of Sets in Venn Diagram
Discover the concept of the complement of the intersection of sets in Venn Diagrams, a crucial element for students and teachers in understanding set theory. This principle highlights the elements that are not part of the intersection of two sets, offering a visual and intuitive method to grasp set relationships. By learning this concept, educators can enhance their teaching strategies, enabling students to better understand complex set operations through visual representation.
 In a diagram of sets A and B, the complement of A ∩ B includes all elements outside the overlapping area. This illustrates elements not shared by A and B.
 With the universal set U containing all possible elements, the complement of A ∩ B is represented by all areas in U not in A ∩ B , showcasing noncommon elements.
 In a Venn Diagram with sets A, B, and C, the complement of A ∩ B ∩ C shows elements in U excluding those common to all three sets, emphasizing unique elements outside these intersections.
 Considering sets of ‘red items’ and ’round items’, the complement of their intersection excludes items that are both red and round, helping students identify items that do not possess both properties simultaneously.
 This concept is used in probability to determine the likelihood of events not occurring together, enhancing students’ understanding of probability theory through visual aids.
What Is a Venn Diagram in Math?
A Venn Diagram is a visual tool used in math to show the relationships between different sets through overlapping circles.
How Do You Read a Venn Diagram?
To read a Venn Diagram, identify each circle as a set and where they overlap, indicating shared elements or characteristics.
Why Are They Called Venn Diagrams?
They are named after John Venn, a mathematician who popularized these diagrams in the 1880s to illustrate logical relationships.
What Is the Middle of a Venn Diagram Called?
The middle, where two or more circles overlap, is called the intersection, representing elements common to all overlapping sets.
Does a Venn Diagram Always Use 2 or 3 Circles?
No, a Venn Diagram can use any number of circles, though 2 or 3 are most common for simplicity and clarity in representation.
Venn Diagrams are indispensable tools in education, offering a visual representation of set relationships that enhance comprehension and analytical skills. By simplifying complex concepts, they become accessible, engaging, and insightful for students, making them a favored resource among educators dedicated to fostering a deeper understanding of mathematics and logic.
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Venn Diagram
Venn Diagrams are used for the visual representation of relationships as they provide a clear, visual method for showing how different sets intersect, overlap, or remain distinct. They are essential tools in mathematics and logic for illustrating the relationships between sets. By employing intersecting circles or other shapes, Venn diagrams visually represent how different sets overlap, share common elements, or remain distinct.
Venn diagrams are commonly used to categorize items graphically, underscoring their similarities and distinctions. In this article, we will discuss all the topics related to the Venn diagram and its definition and examples, and various terms related to it.
 What is a Venn Diagram?
Venn Diagrams are used to represent the groups of data in circles, if the circles are overlapping, some elements in the groups are common, if they are not overlapping, there is nothing common between the groups or sets of data.
A Venn diagram is a graphical representation of mathematical or logical relationships between different sets. It consists of overlapping circles, each representing a set, with the overlapping areas illustrating the common elements shared by the sets.
Venn Diagrams are used for Visual Representation of Relationships as they provide a clear, visual method for showing how different sets intersect, overlap, or remain distinct. This helps in understanding complex relationships between sets in mathematics.
In a Venn diagram sets are represented as circles and the circles are shown inside the rectangle which represents the Universal set. A universal set contains all the circles since it has all the elements present involving all the sets.
Table of Content
Venn Diagram Examples
How to draw a venn diagram, venn diagram for sets operations, terms related to venn diagram, venn diagram symbols, types of venn diagrams, venn diagram for three sets, venn diagram formula, uses and applications of venn diagram, solved example problems on venn diagram, venn diagrams practice questions.
Venn diagrams are highly useful in solving problems of sets and other problems. They are useful in representing the data in picture form. Let’s learn more about the Venn diagram through an example,
Example 1: Take a set A representing even numbers up to 10 and another set B representing natural numbers less than 5 then their interaction is represented using the Venn diagram.
The above symbols are used while drawing and showing the relationship among sets. In order to draw a Venn diagram.
Step 1: Start by drawing a Rectangle showing the Universal Set.
Step 2: According to the number of sets given and the relationship between/among them, draw different circles representing different Sets.
Step 3: Find the intersection or union of the set using the condition given.
Read More: Representation of a Set
There are different operations that can be done on sets in order to find the possible unknown parameter, for example, if two sets have something in common, their intersection is possible. The basic operations performed on the set are,
 Union of Set
 Intersection of Set
 Complement of Set
 Difference of Set
Let’s look at these set operations and how they look on the Venn diagram.
Venn Diagram of Union of Sets
The Union of two or more two sets represents the data of the sets without repeating the same data more than once, it is shown with the symbol ⇢∪.
n(A∪ B) = {a: a∈ A OR a∈ B}
Venn Diagram of Intersection of Sets
The intersection of two or more two sets means extracting only the amount of data that is common between/among the sets. The symbol used for the intersection⇢ ∩.
n(A∩ B)= {a: a∈ A and a∈ B}
Venn Diagram of Complement of a Set
Complementing a set means finding the value of the data present in the Universal set other than the data of the set.
n(A’) = U n(A)
Venn Diagram of Difference of Set
Suppose we take two sets, Set A and Set B then their difference is given as A – B. This difference represents all the values of set A which are not present in set B.
For example, if we take Set A = {1, 2, 3, 4, 5, 6} and set B = {2, 4, 6, 8} then A B = {1, 3, 5}.
In the Venn diagram, we represent the A – B as the area of set A which is not intersecting with set B.
The concept of the Venn diagram is very useful for solving a variety of problems in Mathematics and others. To understand more about it lets learn some important terms related to it.
Universal Set
Universal Set is a large set that contains all the sets which we are considering in a particular situation.
For example, suppose we are considering the set of Honda cars in a society say set A, and let set B is the group of red car in the same society then the set of all the cars in that society is the universal set as it contains the values of both the sets , set A and set B in consideration.
The image representing the Universal set is discussed below,
Subset is actually a set of values that is contained inside another set i.e. we can say that set B is the subset of set A if all the values of set B are contained in set A.
For example, if we take N as the set of all the natural numbers and W as the set of all whole numbers then,
 N = Set of all Natural Numbers
 W = Set of all Whole Numbers
We can say that N is a subset of W all the values of set N are contained in set W i.e.,N ⊆ W
We use Venn diagrams to easily represent a subset of a set. The images discussing the subset of a set are given below,
In order to draw a Venn diagram, first, understand the type of symbols used in sets. Sets can be easily represented on the Venn diagram and the parameters are easily taken out from the diagram itself. We use various types of symbols in drawing Venn diagrams, some of the most important types of symbols used in drawing Venn diagrams are,
Venn Diagram Symbols  Name of Symbol  Description 

∪  Union Symbol  Union symbol is used for taking the union of two or more sets. 
∩  Intersection Symbol  Intersection symbol is used for taking the intersection of two or more sets. 
A’ or A  Compliment Symbol  Complement symbol is used for taking the complement of a set. 
There are various types of Venn diagrams that are widely used in Mathematics and other related fields. The are categorized based on the number of sets involved or circles involved in the Universal set.
 Twoset Venn diagram
 Threeset Venn diagram
 Fourset Venn diagram
 Fiveset Venn diagram
We can represent three sets easily using the Venn Diagram. Their representation is done by three overlapping circles. Suppose we take three sets of Set A of the people who play cricket. Set B of the people who are graduates and Set C of the people who are 18 years and above of the age.
Then the Venn diagram representing the above three sets is drawn using three circles and taking their intersection wherever required.
We can represent the intersection of three sets using the Venn diagram. The below image represents the intersection of three sets.
We can find the various parameters using the above Venn diagram.
Suppose we have to find,
 No of graduates who play cricket it is given by B⋂C
 No of graduates who play cricket and are at least 18 years old is given by A⋂B⋂C , etc.
Also Check:
Difference of Sets Universal Set Equal Sets
We use various formulas of the set to find various parameters of the sets.
Let’s take two sets, set A and set B then the various formulas of the sets are,
n(A U B) = n(A) + n(B) – n (A ⋂ B)
 n(A) represents the number of elements in set A,
 n(B) represents the number of elements in set B,
 n(A U B) represent the number of elements in A U B, and
 n(A ⋂ B) represent the number of elements in A ⋂ B
Similarly, for three sets, Set A, Set B, and Set C we get,
n (A U B U C) = n(A) + n(B) + n(C) – n(A ⋂ B) – n(B ⋂ C) – n(C ⋂ A) + n(A ⋂ B ⋂ C)
We can understand these formulas with the help of the example discussed below,
Example: In a class of 40 students, 18 like Mathematics, 16 like Science, and 10 like both Mathematics and Science. Then find the students who like either Mathematics or Science.
Let A be the set of students who like Mathematics and B be the set of students who like Science, then n(A) = 18, n(B) = 16, and n(A ⋂ B) = 10 Now to find the number of students who like either Mathematics or Science i.e. n(A U B) we use the above formula. n(A U B) = n(A) + n(B) – n (A ⋂ B) ⇒ n(A U B) = 18 + 16 – 10 ⇒ n(A U B) = 24
Venn diagrams have various use cases such as solving various problems and representing the data in an easytounderstand format. Various applications of Venn Diagrams are:
 The relation between various sets and their operations can be easily achieved using Venn diagrams.
 They are used for explaining large data sets in a very easy way.
 They are used for logic building and finding the solution to complex data problems.
 They are used to solve problems based on various analogies.
 Analysts use Venn diagrams to represent complex data in easily understandable ways, etc.
Related Article on Venn Diagram:
Operations on Sets Types Of Sets Set Theory Formulas
Example 1: Set A= {1, 2, 3, 4, 5} and U= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.
Represent a’ or a c on the venn diagram..
Venn Diagram for A’
Example 2: In a Group of people, 50 people either speak Hindi or English, 10 prefer speaking both Hindi and English, 20 prefer only English. How many people prefer speaking Hindi? Explain both by formula and by Venn diagram.
According to formula, n(H∪E) = n(H) + n(E) – n(H∩E) Both English and Hindi speakers, n(H∩E) = 10 English speakers, n(E)= 20 Either Hindi or English, n(H∪E)= 50 50= 20+ n(H) – 10 n(H)= 50 – 10 n(H)= 40 From Venn Diagram,
Example 3: In a Class, Students like to play these games Football, Cricket, and Volleyball. 5 students Play all 3 games, 20 play Football, 30 play Volleyball, and 40 play Cricket. 10 play both cricket and volleyball, 12 play both football and cricket, 9 play both football and volleyball. How many students are present in the class?
n(F∪ C∪ V)= n(F)+ n(C)+ n(V) – n(F∩C) – n(F∩V) – n(C∩V)+ n(F∩ C∩ V) n(F∪ C∪ V)= 20+ 30+ 40 10129+5 n(F∪ C∪ V)= 64 There are 64 Students in the class.
Example 4: Represent the above information with the help of a Venn diagram showing the amount of data present in each set.
Above information should look something like this on Venn diagram,
Example 5: Below given Venn diagram has all the sufficient information required to show the data of all the sets possible. Observe the diagram carefully then answer the following.
 What is the value of n(A∩ B∩ C)?
 What is the value of n(C)?
 What is the value of n(B ∩ A)?
 What is the value of n(A∪ B∪ C)?
 What is the value of n(B’)?
Observing the Venn diagram, the above questions can be easily answered, 1. n(A ∩ B∩ C)= 5 2. n(C)= 15+ 5+5+5= 30 3. n( B∩A)= 5+5= 10 4. n(A∪ B∪ C)= 15+ 20+ 10+ 5+ 5+ 5+ 5= 65 5. n(B’)= U n(B)= 100 (20+ 5+ 5+ 5)= 100 35= 65
Q1. Consider two sets, A and B, where A represents fruits and B represents vegetables. Set A contains apples, bananas, and grapes, while set B contains carrots, lettuce, and apples. Draw a Venn diagram to represent these sets. How many items are only in the fruit category?
Q2. In a small neighborhood, 10 households have dogs, 7 have cats, and 3 households have both dogs and cats. How many households have at least one kind of pet? Draw a Venn diagram to represent this situation.
Q3. In a sports club, 120 members play tennis, 150 play badminton, and 50 play both tennis and badminton. How many members play either tennis or badminton? Create a Venn diagram to help you answer.
Venn diagrams, created by English logician John Venn in the 1880s. Venn diagrams are a powerful tool for visualizing the relationships between different sets, making complex concepts more accessible and easier to understand. Using overlapping circles within a rectangle (the universal set), they illustrate how sets intersect, differ, and relate, with each circle representing a different set. Overlapping regions show common elements, while nonoverlapping areas highlight unique element. Venn diagrams are applied across various fields for problemsolving, data presentation, and logical reasoning , making them a versatile tool for educators, students, and professionals alike.
Venn Diagrams – FAQs
What is a venn diagram in mathematics.
Venn diagrams are important ways to represent complex logical relations. They were first implemented by the famous mathematician John Venn. They are used to represent the relation between various sets.
How to Read a Venn Diagram?
We can read the Venn diagram with the help of the following steps, Observe all the circles in the entire diagram as they represent various sets of data. Every circle in the diagram represent a particular data set. These circles are overlapped according to various conditions present. Study the circles and their interaction to identify various data in the diagram
What is A ∩ B Venn Diagram?
A ∩ B signifies the common element between set A and set b and it is read as A intersection B. In Venn, diagram set A is represented using a circle and similarly set b is represented using another circle then their intersection A ∩ B is represented by the overlapping of the circle of set A and set B.
What is ∩ in a Venn Diagram?
In a Venn Diagram, the symbol ∩ represents the intersection, which indicates the portion that is common to both sets.
What are Types of Venn Diagram?
Venn diagrams come in various types based on the number of sets they represent. The different types include: Twoset Venn diagram Threeset Venn diagram Fourset Venn diagram Fiveset Venn diagram
What is Venn Diagram is Used for?
A Venn diagram is a visual tool used in logic theory and set theory to show the relationship between different sets or data.
How to Use a Venn Diagram?
To use a Venn Diagram follow the steps added below: Step 1: Draw circles to represent each set. Step 2: Label each circle with the name of the set. Step 3: Place common elements in the overlapping areas. Step 4: Place unique elements in the nonoverlapping areas.
What are the Parts of a Venn Diagram?
The main parts of a Venn Diagram are: Circles: Representing the sets. Overlapping Areas: Showing common elements. Nonoverlapping Areas: Showing unique elements. Universal Set: Often represented by a rectangle surrounding the circles.
What is the Purpose of a Venn Diagram?
The purpose of a Venn Diagram is to visually display the relationships between different sets, making it easier to compare and contrast data, identify similarities and differences, and understand set operations.
What is Intersection in a Venn Diagram?
Intersection in a Venn Diagram is the overlapping area of the circles, showing elements that are common to all the sets involved.
What is Union in a Venn Diagram?
Union in a Venn Diagram includes all elements from all the sets, represented by the entire area covered by the circles.
How can Venn Diagrams Help in ProblemSolving?
Venn Diagrams help in problemsolving by providing a clear visual representation of sets and their relationships, which makes it easier to analyze complex data and find solutions.
What are Some Common Uses of Venn Diagrams?
Common uses of Venn Diagrams include: Comparing and contrasting information. Solving math problems involving sets. Analyzing survey data. Illustrating logical relationships in various fields.
Can Venn Diagrams be used for more than Three Sets?
Yes, Venn Diagrams can be used for more than three sets, but they become more complex and harder to draw. Typically, diagrams with up to three sets are most common and easiest to understand.
How do you Draw a Venn Diagram for Three Sets?
To draw a Venn Diagram for three sets follow the steps added below: Step 1: Draw three overlapping circles. Step 2: Label each circle with the set name. Step 3: Fill in the overlapping areas with common elements. Step 4: Place unique elements in the nonoverlapping parts of each circle.
Why are Venn Diagrams Important in Statistics?
Venn Diagrams are important in statistics because they help visualize relationships between different data sets, making it easier to analyze and interpret statistical data.
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How to Solve Venn Diagrams with 3 Circles
Venn diagrams with 3 circles: video lesson, what is the purpose of venn diagrams.
A Venn diagram is a type of graphical organizer which can be used to display similarities and differences between two or more sets. Circles are used to represent each set and any properties in common to both sets will be written in the overlap of the circles. Any property unique to a particular set is written in that circle alone.
For example, here is a Venn diagram comparing and contrasting dogs and cats.
The Venn diagram shows the following information:
 Have nonretractable claws
 Have round pupils
 Roam the street
 Have retractable claws
 Have slit pupils
Both dogs and cats:
 Can be pets
 Have 4 legs
A Venn diagram with three circles is called a triple Venn diagram.
A Venn diagram with three circles is used to compare and contract three categories. Each circle represents a different category with the overlapping regions used to represent properties that are shared between the three categories.
For example, a triple Venn diagram with 3 circles is used to compare dogs, cats and birds.
Dogs, cats and birds can all have claws and can also be pets.
Only birds:
 Have a beak
 Have 2 legs
Only both dogs and cats:
Only both dogs and birds:
Only both cats and birds:
 Don’t need walks
How to Make a Venn Diagram with 3 Circles
 Write the number of items belonging to all three sets in the central overlapping region.
 Write the remaining number of items belonging each pair of the sets in their overlapping regions.
 Write the remaining number of items belonging to each individual set in the nonoverlapping region of each circle.
Make a Venn Diagram for the following situation:
30 students were asked which sports they play.
 20 play basketball in total
 16 play football in total
 15 play tennis in total
 10 play basketball and tennis
 11 play basketball and football
 9 play football and tennis
 7 play all three
 Write the number of items belonging to all three sets in the central overlapping region
When making a Venn diagram, it is important to complete any overlapping regions first.
In this example, we start with the students that play all three sports. 7 students play all three sports.
The number 7 is placed in the overlap of all 3 circles. The shaded region shown is the overlapping area of all three circles.
2. Write the remaining number of items belonging each pair of the sets in their overlapping regions
There are 3 regions in which exactly two circles overlap.
There is the overlap of basketball and tennis, basketball and football and then tennis and football.
There are 10 students that play both basketball and tennis. The overlapping region of these two circles is shown below. We already have the 7 students that play all three sports in this region.
Therefore we only need 3 more students who play basketball and tennis but do not play football to make the total of this region add up to 10.
The next overlapping region of two circles is those that play basketball and football. There are 11 students in total that play both.
The overlapping region of the basketball and football circles is shown below.
There are already 7 students who play all three sports and so, a further 4 students must play both basketball and football but not tennis in order to make the total in this shaded region add up to 11 students.
The next overlapping region of two circles is those that play football and tennis. There are 9 students in total that play both.
The overlapping region of the football and tennis circles is shown below.
There are already 7 students who play all three sports and so, a further 2 students must play both football and tennis but not basketball in order to make the total in this shaded region add up to 9 students.
Write the remaining number of items belonging to each individual set in the nonoverlapping region of each circle
There are three individual sets which are represented by the three circles. There are those that play basketball, football and tennis.
20 students play basketball in total. These 20 students are shown by the shaded circle below.
We already have 3, 7 and 4 students in the overlapping regions. This is a total of 14 students so far. We need a further 6 students who only play basketball in order for the numbers in this circle to make a total of 20.
The next individual sport is football. 16 students play football in total.
There are already 4, 7 and 2 students in the overlapping regions. This makes a total of 13 students so far.
3 more students are required to make the circle total up to 16. 3 students play only football and not basketball and tennis.
Finally, there are 15 students who play tennis shown by the shaded region below.
There are already 3, 7 and 2 students in the overlapping regions, making a total of 12 students.
A further 3 students are required to make the total of 15 students in this circle.
3 students play tennis but not basketball or football.
The values in each circle sum to 28 students.
That is 6 + 4 + 3 + 7 + 3 + 2 + 3 = 28.
Since there are 30 students who were asked in total, a further 2 students must play none of these three sports.
How to Solve a Venn Diagram with 3 Circles
To solve a Venn diagram with 3 circles, start by entering the number of items in common to all three sets of data. Then enter the remaining number of items in the overlapping region of each pair of sets. Enter the remaining number of items in each individual set. Finally, use any known totals to find missing numbers.
Venn diagrams are particularly useful for solving word problems in which a list of information is given about different categories. Numbers are placed in each region representing each statement.
100 people were asked which pets they have.
 32 people in total have a cat
 18 people in total have a rabbit
 10 people have just a dog and a rabbit
 21 people have just a dog and a cat
 7 people have just a cat and a rabbit
 3 people own all three pets
How many people just have a dog?
Start by entering the number of items in common to all three sets of data
3 people own all three pets and so, a number 3 is written in the overlapping region of all three circles.
Then enter the remaining number of items in the overlapping region of each pair of sets
10 people have just a dog and a rabbit.
Since 3 people are already in this region, 7 more people are needed.
21 people have just a dog and a cat.
Since 3 people are already in this region, 18 more people are needed.
7 people have just a cat and a rabbit.
Since 3 people are already in this region, 4 more people are needed.
Enter the remaining number of items in each individual set
32 people in total have a cat.
There are already 18 + 3 + 4 = 25 people in this circle.
Therefore a further 7 people are needed in this circle to make 32.
7 people just own a cat and no other pet.
18 people in total have a rabbit.
There are already 7 + 3 + 4 = 14 people in this circle.
Therefore a further 4 people are needed in this circle to make 18.
4 people just own a rabbit and no other pet.
Finally, use any known totals to find missing numbers
We are now told that 25 people own none of these pets. This means that a 25 is written outside of all of the circles but still within the Venn diagram.
The question requires the number of people who just own a dog.
There are 100 people in total and so, all of the numbers in the complete Venn diagram must add up to 100.
Adding the numbers so far, 3 + 7 + 4 + 18 + 4 + 7 + 25 = 68 people in total.
Since the numbers must add to 100, there must be a further 32 people who own a dog.
Now all of the numbers in the Venn diagram add to 100.
Venn Diagram with 3 Circles Template
Here is a downloadable template for a blank Venn Diagram with 3 circles.
How to Shade a Venn Diagram with 3 Circles
Here are some examples of shading Venn diagrams with 3 sets:
Shaded Region: A
Shaded Region: B
Shaded Region: C
Shaded Region: A∪B
Shaded Region: B∪C
Shaded Region: A∪C
Shaded Region: A∩B
Shaded Region: B∩C
Shaded Region: A∩C
Shaded Region: A∪B∪C
Shaded Region: A∩B∩C
Shaded Region: (A∩B)∪(A∩C)
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Mathematicians Discover New Shapes to Solve DecadesOld Geometry Problem
September 20, 2024
These three objects have constant width, meaning that when placed between two flat surfaces, they roll smoothly, as if they were balls — even though it doesn’t look like they should be able to.
Christopher Webb Young/ Quanta Magazine
Introduction
In 1986, after the space shuttle Challenger exploded 73 seconds into its flight, the eminent physicist Richard Feynman was called in to find out what had gone wrong . He later demonstrated that the “Oring” seals, which were meant to join sections of the shuttle’s solid rocket boosters, had failed due to cold temperatures, with catastrophic results. But he also discovered more than a few other missteps.
Among them was the way NASA had calculated the Orings’ shape. During preflight testing, the agency’s engineers had repeatedly measured the width of the seals to verify that they had not become distorted. They reasoned that if an Oring had been slightly squashed — had become, say, an oval, instead of maintaining its circular shape — then it would no longer have the same diameter all the way around.
These measurements, Feynman later wrote, were useless. Even if the engineers had taken an infinite number of measurements and found the diameter to be exactly the same each time, there are many “bodies of constant width,” as these shapes are called. Only one is a circle.
Arguably the best known noncircular body of constant width is the Reuleaux triangle, which you can construct by taking the central region of overlap in a threecircle Venn diagram. For a given width in two dimensions, a Reuleaux triangle is the constantwidth shape with the smallest possible area. A circle has the largest.
In three dimensions, the largest body of constant width is a ball. In higher dimensions, it’s simply a higherdimensional ball — the shape swept out if you hold a needle at a point and let it rotate freely in every direction.
Mark Belan for Quanta Magazine
But mathematicians have long wondered if it’s always possible to find smaller constantwidth shapes in higher dimensions. Such shapes exist in three dimensions: Though these Reuleauxlike blobs might look a bit pointy, sandwich them between two parallel planes and they will roll smoothly, like a ball. But it’s much harder to tell whether this is true in general. It could be that in higher dimensions, the ball is optimal. And so in 1988, Oded Schramm, then a graduate student at Princeton University, asked a simplesounding question: Can you construct a constantwidth body in any dimension that is exponentially smaller than the ball?
Now, in a paper posted online in May, five researchers — four of whom grew up in Ukraine and have known each other since their high school or college days — have reported that the answer is yes.
The result not only solves a decadesold problem, but gives mathematicians their first glimpse into what these mysterious higherdimensional shapes might look like. Although these shapes are easy to define, they’re surprisingly mysterious, said Shiri Artstein , a mathematician at Tel Aviv University who wasn’t involved in the work. “Any new thing we learn about them, any new construction or computation, is at this point interesting.” Now researchers can finally access a corner of the geometric universe that was once completely unapproachable.
Planting the Seed
Andrii Arman and Danylo Radchenko met in the mid2000s at a mathfocused high school in Kyiv and were also teammates on Ukraine’s competitive Math Olympiad squad. They became friends, but didn’t stay in close touch. When their mathematical work later pulled them independently into the orbits of both Andriy Prymak and Andrii Bondarenko — who had attended Kyiv National University together in the 1990s — they reconnected. The four mathematicians have since moved to different cities around the world and pursued different research programs, but they gather twice a week over Zoom to work together on tough geometric proofs.
With their collaborators, Andrii Bondarenko (left) and Danylo Radchenko recently proved that you can always find small shapes of constant width in high dimensions.
From left: Ekaterina Poliakova/Norwegion University of Science and Technology; Grégory Hau
Constantwidth shapes were not initially on the agenda. Last year, the group was instead trying to answer a related question called the Borsuk problem, which had stumped prominent mathematicians for over a century. But an idea kept popping up during their meetings: When Schramm posed his question about constantwidth bodies in the 1980s, he also suggested that understanding such shapes might provide a way to tackle the Borsuk problem.
The Ukrainian mathematicians had been pursuing a different approach, and some of them were reluctant to change focus. But Bondarenko, now at the Norwegian University of Science and Technology, insisted that they try, even if it didn’t help them directly. “He was always emphasizing that the problem is important in its own right,” said Arman, who is currently a postdoctoral researcher at the University of Manitoba. Eventually, the rest of the team agreed to make the attempt.
To understand what they did, it helps to think back to the Reuleaux triangle in two dimensions. Say you want to build a Reuleaux triangle of a given width. First draw an equilateral triangle — what the mathematicians call a seed. Choose a point on the triangle’s boundary and draw a circle around it with a radius equal to the width you want the final shape to be. Now do this at every point on the triangle’s boundary, so that you get a set of infinitely many circles.
Look at the region where those circles overlap. Somewhere within it, you’ll be able to find a body of constant width — you just have to figure out which subset of your seed you actually need. In this case, you can look at just the three vertices of the equilateral triangle, rather than all the points on its boundary. Draw circles around those three points, and you’ll get a Venn diagram; its overlapping region is the Reuleaux triangle.
In higher dimensions, it’s possible to use the same approach. Start with a set of points: your seed. Draw a ball around each point, take their intersection, and look for the body of constant width that lives inside that new space. But it’s much harder in high dimensions to figure out what subset of your seed will give you the shape you want.
Arman, Bondarenko, Prymak and Radchenko experimented with different seeds and eventually came up with a particular curve they wanted to use. They knew that this curve would give them a region that contained a sufficiently small body of constant width. But they wanted to understand what the constantwidth body itself would look like. As they searched for an answer, Arman came across a post from 2022 on the questionandanswer site MathOverflow. The poster, Fedor Nazarov of Kent State University, had been independently trying to answer Schramm’s question, and his approach looked remarkably similar to the Ukrainian team’s, though he’d gotten stuck. The quartet invited him to join them. It was then that Nazarov realized something that the rest of them had missed: The shape their seed gave them did not just contain a constantwidth body. It was one.
Andrii Arman (left) and Andriy Prymak comprise half of a fourperson team of mathematicians from Ukraine who have been collaborating for years.
Jaskaran Singh
Their work provides a surprisingly simple algorithm for building an n dimensional shape of constant width whose volume is at most 0.9 n times that of the ball. That limit is, in a sense, arbitrary, Arman said. It should be possible to find even smaller bodies of constant width. But it is enough to answer Schramm’s question, proving that as the number of dimensions increases, the gap between the volumes of the smallest and largest constantwidth bodies grows exponentially. Despite the complex ideas behind their result, Arman said, their construction is something undergraduates should be able to verify.
For Gil Kalai of Hebrew University, there is personal satisfaction in seeing an answer for Schramm, his former student, who died in 2008 in a hiking accident after making significant advances on questions in many different fields. But Kalai is also excited to explore the theoretical consequences of the result. Previously, he said, it was possible that in higher dimensions, these shapes would all simply behave like balls, at least when it came to the property of volume. But “this is not the case. So this means that the theory of these bodies in high dimensions is very rich,” he said.
That theory might even have applications. In lower dimensions, after all, bodies of constant width are already surprisingly useful: The Reuleaux triangle, for example, shows up in the form of drill bits and guitar picks and tamperproof nuts for fire hydrants. According to Arman, in higher dimensions, their new shapes might be useful in the development of machine learning methods for analyzing highdimensional data sets. Bondarenko — known within the group for what Arman calls his “crazy ideas” — has also proposed connections to distant branches of mathematics.
The search for the smallest possible body of constant width — which remains open in all dimensions greater than 2 — continues. The group briefly used their construction to investigate one promising candidate in three dimensions, but it let them down: It turned out to be a tiny fraction of a percent larger than the smallest known body. For now, the mathematicians have decided to give up the chase and return to their work on Borsuk’s problem. In their wake, they’ve left behind a world of new highdimensional shapes for others to explore.
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This article will look at the types of Venn diagram questions that might be encountered at middle school and high school, with a focus on exam style example questions and preparing for standardized tests. We will also cover problemsolving questions. Each question is followed by a worked solution. How to solve Venn diagram questions
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Arguably the best known noncircular body of constant width is the Reuleaux triangle, which you can construct by taking the central region of overlap in a threecircle Venn diagram. For a given width in two dimensions, a Reuleaux triangle is the constantwidth shape with the smallest possible area. A circle has the largest.