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Maths assignment class viii | quadrilateral, math assignment class xi ch - 2 | relations & functions, mathematics assignment chapter 2 class -11 relations and functions.

Extra questions of chapter 2 class 11  with answer and  hints to the difficult questions. Important and useful math. assignment for the students of class 11

ASSIGNMENT ON RELATIONS & FUNCTIONS

i)  Find x and y if  (x + 3, 5) = (6, 2x + y)        

Ans (x = 3, y = - 1)

ii) If ordered pair (x, -1) and (5, y) belongs to the set {(a, b): b = 2a - 3}, find the value of x and y

[Ans a = 2, b = 1]

(iv) If a  ∈  {-1, 2, 3, 4, 5}, b  ∈  {0, 3, 6}, write the set of all ordered pairs (a, b) such that a + b = 5 Solution -1 + 6 = 5  ⇒  (-1, 6)  ∈  (a, b) 2 + 3 = 5  ⇒  (2, 3)  ∈  (a, b) 5 + 0 = 5  ⇒  (5, 0)  ∈  (a, b) ⇒  R = {(-1, 6), (2, 3), (5, 0)}

i)  If A = {1, 3, 5, 6} and B = {2, 4}, find A  ×  B and B  ×  A

ii) If A = {1, 2, 3}, B = {3, 4} and C ={1, 3, 5}, find 

(a) A × (B ∩ C)     

(b)  (A  ×  B) ∩ (A  ×  C)

iii) If A = {1, 3, 5} , B = {x, y} then represent A  ×  B and B  ×  A in arrow diagrams

Find the domain and range of the relation R defined by   R = {(x,  x 3  ) : x is a prime number less than 10}              

(i) n(A) = 3, n(B) = 4, then find  n(A  ×  A  ×  B)  

Ans: 36 

Ans : {(1, 4), (4, 4)} 

[Ans:    2 2 ✕ 2   =  2 4  =16 ]

 R = {(x, y) : |x - y|  is odd}. Write R in roster form.

Ans:  R = {(1, 4), (1, 6),  (2, 9), (3, 4), (3, 6), (5, 4), (5, 6)

Express the following functions as set of ordered pair and determine their ranges

(i)   f : A ➝  R, f(x) = x 2  + 1,  where A = {-1, 0, 2, 4}   

Ans(i): Range = {1, 2, 5, 17}       

  Ans(ii) : Range = {2, 4, 6, 8, 10, 12, 14}

Let f : R - {2}  ➝   R be defined by 

 and g : R  ➝  R be  defined by g(x) = x + 2.  Find whether f = g or not.

Solution:     

Also g(x) = x + 2   ⇒  f(x) = g(x)

But  Domain of f(x) = R - {2}  and the domain of g(x) = R  and two functions 

are equal only if their domains are equal.

Hence f(x) ≠ g(x)

Find the domain for which the functions f(x) =  2x 2  - 1  and g(x) = 1 - 3x are equal

Solution:  

Here f(x) = g(x)

2x 2   - 1 = 1 - 3x   ⇒   2x 2  + 3x – 2 = 0  

⇒  x = -2, 1/2

⇒  f(x) and g(x) are equal on the set  {-2, 1 / 2 }

 If f, g, h are three functions defined from R to R as follows

(i) f(x) =  x 2       (ii)  f(x) = sinx    (iii) f(x) =  x 2  + 1

Find the range of each function

(i)   For all values of x,  f(x) takes only   +ve value. Also if x = 0, then f(x) = 0 . So Range  of  f(x)  =  [0, ∞)

(ii) Since  -1 ≤ sinx ≤ 1 for all values of x. So Range of f(x) = [-1,1]

(iii) Since   x 2  is ≥ 0  ⇒  x 2  + 1 ≥ 1   ⇒  Range of f(x) = [1, ∞)

Solution: 

Find the domain of the function f(x) defined by  

f(x) is defined for all x satisfying

4 - x ≥ 0 and x 2  - 1 > 0

x  ∈  (- ∞, -1) U (1, ∞),   But   x ≤ 4

Therefore : D f   = (- ∞, -1) U (1, 4]

By using quadratic formula we get 

Now x is defined (real) if 1 - 4y 2  ≥ 0 and y ≠ 0

⇒   4y 2  - 1  ≤ 0     ⇒   (2y + 1)(2y - 1)  ≤ 0

Critical points are  y = -1/2 and  1/2

(2y + 1)(2y - 1)  ≤ 0   and  y ≠ 0  ⇒  y   ∈  [ -1/2, 1/2] - {0}

⇒ R f  = [ -1/2, 1/2] - {0}   

Solution Hint: 

For range find x in terms of y we get

Solution (iii)

f (x) is defined for all real numbers except at x = 3

Therefore D f  = R - 3

 But x = 3  ∉  D f    ⇒   y = 3 + 3 = 6  ∉  R f     ⇒  R f  = R - {6}

Find the domain and range of the following functions

i) f(x) = |x - 1|        Ans : [D = R]  R = [0, ∞)

Ans: Domain = R - {2},   Range = R - {-1}

Ans: Domain = [-4, 4], Range = [0, 4]

Ans: Domain = (-3, 3) Range = [1, ∞)

Solution (vi)

⇒  9y 2  - x 2 y 2  = 9

⇒ x 2 y 2  = 9y 2  - 9

  x is defined if  (y + 1)(y - 1) ≥ 0 and  y ≠ 0

x is defined if  y ≤  -1,   y ≥ 1 and  y ≠ 0

⇒   y ∈ (-∞, -1] ⋃ [1, ∞) ⇒ R f   = (-∞, -1] ⋃ [1, ∞)

Ans: D f   = (-∞, -3] ⋃ (-1, 3]

Draw the graph of the following write its range

Find the domain of the following function

Solution: Let |x| = y 

 Critical points are  y = 5, y = 3

y = |x| = 5  and  y = |x| = 3  ⇒  x = 土 5,  x =  土 3

Question 20 : (DAV SP 2023)

Find the inverse relation R -1  in each of the following

(i) R = {(1,2), (1,3), (2, 3), (3, 2), (5, 6)          

Ans {(2,1), (3,1), (3,2), (2,3), (6,5)}

(ii) R = {(x, y) : x, y  ∈  N, x + 2y = 8 

Ans {(3, 2), (2, 4), (1,6)

Solution If  x = 2, y = 3  ⇒  2 + 2  ×  3 = 8  ⇒  (2, 3)  ∈  R

If  x = 4, y = 2  ⇒  4 + 2  ×  2 = 8  ⇒  (4, 2)  ∈  R

If  x = 6, y = 1  ⇒  6 + 2  ×  1 = 8  ⇒  (6, 1)  ∈  R

R = {(2, 3), (4, 2), (6, 1)}   ⇒  {(3, 2), (2, 4), (1, 6)}

Let f : R ➝ R be a function given by f(x) =  x 2  + 1. Find     f  -1 {10, 37}   

If f(x) = y then  x = f  -1 (y)

f  -1 (10) = x  ⇒  f(x) = 10  ⇒  x 2  + 1 = 10 

⇒  x 2  = 9   ⇒  x =  土  3

 f  -1 (37) = x  ⇒  f(x) = 37  ⇒  x 2  + 1 = 37 

⇒  x 2  = 36   ⇒  x =  土  6

 f  -1 {10, 37} = {-3, 3, - 6, 6}

 Let A = {-2, -1, 0, 1, 2} and f : A  ➝  Z be a function defined by  f(x) = x 2  - 2x - 3. Find

(i) Range of f i.e. f(A)          (ii) Pre - image of 6, - 3 and 5

(i)  f(A) = {f(-2), f(-1), f(0), f(1), f(2) } = {5, 0, - 3, - 4, - 3} = {- 4,- 3, 0, 5}

(ii) Let Pre - image of 6 = x  ⇒    f  -1 (6) = x  ⇒  f(x) = 6

⇒   x 2  - 2x - 3 = 6   ⇒     x 2  - 2x - 9= 0 

There is no real value of x which satisfies this equation. 

So Pre - image of  6 = Φ

Let Pre - image of - 3 = x  ⇒    f  -1 (-3) = x  ⇒  f(x) = - 3

⇒   x 2  - 2x - 3 = - 3   ⇒     x 2  - 2x = 0  ⇒   x = {0, 2}

Let Pre - image of 5 = x  ⇒    f  -1 (5) = x  ⇒  f(x) = 5

⇒   x 2  -2x - 3 = 5   ⇒    x 2  - 2x - 8 = 0  ⇒   x = {-2, 4}

(i) f(x) = 3x 4  - 5x 2  + 9, find  f(x - 1)          

Ans [3x 4  - 12x 3  + 13x 2   - 2x + 7]

 [Ans x = -1, 2/3]

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HSC Assignment 2021 For HSC Exam 2021 Candidates

HSC 2021 Exam, Class 11 Assignment 2021 & 2022 for College, HSC, Intermediate 1st year students has been published. If you are a candidate of class 11 and going to take part HSC exam 2021 & 2022, then you need to get Class 11 or HSC assignment 2021 questions & solutions. So here you can get your XI class assignment solves. একাদশ শ্রেণীর এসাইনমেন্ট, এইচএসসি ২০২১ & ২০২২ পরীক্ষার্থীদের এসাইনমেন্ট উত্তর। নিচে একাদশ শ্রেণীর এইচএসসি ২০২১ & ২০২২ প্রথম সপ্তাহের সকল এসাইনমেন্ট প্রশ্ন দেওয়া আছে।

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Due to the recent corona epidemic, almost all the educational institutions in the country are closed and classes are being conducted online. However, the college authorities are in a dilemma as to how the HSC examination 2022 of class 11 will be conducted. In Higher secondary colleges, instead of annual examinations from class 11 to class 12, initiatives have been taken to improve the next class through assignment. Following this, various government and private colleges of the country have provided class 11 assignments / HSC assignments for their students on their own initiative.

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There are a total of 30 weeks assignments. So each week has some subjects task. Every week Education board will publish of each week’s assignments for HSC 2021 syllabus completion. In this case, we will try to answer all the HSC assignment questions. So that all class 11 students can easily done their assignment task.

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DSHE has published the 4th week HSC assignment questions on their website. So here we are shearing all 4th week HSC assignment for 2022 final board exam candidates. Here you can get the 4th week’s Bangla, Physics, Logic, Accounting, Economics, Civics, Khaddo O Pushti for HSC 2022 exam. 4th Week PDF

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HSC 3rd Week Assignment Question & Answer

DSHE has published the 3rd week HSC assignment questions on their website. So here we are shearing all 2nd week HSC assignment for 2022 final board exam candidates. Here you can get the 3rd week’s Bangla and English for HSC 2022 exam.

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HSC 2nd Week Assignment Question & Answer

DSHE has published the 2nd week HSC assignment questions on their website. So here we are shearing all 2nd week HSC assignment for 2022 final board exam candidates. Here you can get the 2nd week’s English, Physics, Economics, Logic, Civics and Accounting Assignments for HSC 2022 exam.

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You can get 1st week HSC assignment questions & answer on our website.

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HSC Assignment Routine & Syllabus

What can be in the Assignment of HSC 2022?

First of all, the assignments of HSC 2022 have to be presented separately on different subjects. The assigned teacher of the specific subject will select the questions of the HSC 2022 assignment in that subject and provide them to the students. Students will solve the question of HSC assignment on the subject or prepare the assignment and hand it over to the college authorities or teachers.

Syllabus of HSC Class 11 Assignment

The syllabus has been officially given in HSC assignment. A 30-week routine has been given to complete the Class XI syllabus. All colleges have to follow the assignment of this HSC 2022 syllabus. So for the syllabus of HSC assignment one has to collect HSC assignment every week by contacting the college authorities. And the assignment should be answered as per the instructions given in the question of HSC 2022 assignment.

How to submit the assignment of Inter 1st Year?

After the HSC assignment is completed, if it is a handwritten assignment, it should be handed over directly to the college authorities or to the teacher. And if the assignment is in econo digital format, it can be sent via email to the college authorities or to the teacher’s email.

How to do XI Class Students HSC Assignment?

Since this HSC assignment is a bridge for xi class students, the quality of this assignment needs to be improved a bit compared to college students. The cover page of the HSC assignment should be beautiful and is very good if printed in color. It will be decided by the college whether to write the assignment by hand or by typing it on the computer. It is better to use offset press to ensure that the HSC assignment pages are clear and clean. College assignments need to understand the Hawaiian questions very well, to understand what you really want. In case of giving a statement, a clear explanation should be given. Anything unreasonable or something that is not sought in the question cannot be discussed. The assignments of college, hsc, intermediate, class 11, class 12 are more important than the assignments of others so you have to keep that in mind while making the assignment. The language of assignment writing should be pretty clear. One can give one’s own opinion on the subject in question and it is also good to describe any experience.

এইচএসসি ২০২২ অ্যাসাইনমেন্ট ২০২১ এর সমাধান

সম্প্রতি করণা মহামারীর কারণে দেশের প্রায় সকল শিক্ষাপ্রতিষ্ঠান বন্ধ রয়েছে এবং অনলাইনের মাধ্যমে ক্লাস পরিচালনা হচ্ছে । তবে class 11 , class 12 এর সমাপনী পরীক্ষা কিভাবে হবে তা নিয়ে কলেজ কর্তৃপক্ষ দ্বিধাদ্বন্দ্বে রয়েছে । মাধ্যমিক স্কুলে ষষ্ঠ শ্রেণি থেকে নবম শ্রেণী পর্যন্ত বার্ষিক পরীক্ষার বদলে অ্যাসাইনমেন্ট এর মাধ্যমে পরবর্তী শ্রেণীতে উন্নত ও মেধাক্রম নির্ণয়ের উদ্যোগ গ্রহণ করা হয়েছে । এরই ধারাবাহিকতায় দেশের বিভিন্ন সরকারি ও বেসরকারি কলেজ তাদের নিজ উদ্যোগে তাদের ছাত্র-ছাত্রীদের জন্য অ্যাসাইনমেন্ট প্রদান করেছেন।

কলেজ, এইচএসসি, ক্লাস ১১ ও ১২ এর এসাইনমেন্ট এ কি কি থাকতে পারে?

প্রথমত class 11 , class 12 এর অ্যাসাইনমেন্ট গুলো আলাদা আলাদা বিষয়ের উপর আলাদা আলাদা ভাবে উপস্থাপন করতে হবে । নির্দিষ্ট বিষয়ের নির্ধারিত শিক্ষক উক্ত বিষয়ে অ্যাসাইনমেন্ট এর প্রশ্ন নির্বাচন করে ছাত্রছাত্রীদের প্রদান করবেন । ছাত্র-ছাত্রীরা উক্ত বিষয়ে অ্যাসাইনমেন্ট এর প্রশ্ন সমাধান করে বা অ্যাসাইনমেন্ট প্রস্তুত করে কলেজ কর্তৃপক্ষ বা শিক্ষকের নিকট হস্তান্তর করবেন।

একাদশ শ্রেণীর এসাইনমেন্ট এর সিলেবাস

কলেজ অ্যাসাইনমেন্টে সরকারি ভাবে কোন সিলেবাস দেওয়া হয়নি। সকল কলেজ আলাদা আলাদা ভাবে তাদের নিজ উদ্যোগে সিলেবাস তৈরি করেছে । সুতরাং এসাইনমেন্ট এর সিলেবাস এর জন্য কলেজ কর্তৃপক্ষের নিকট যোগাযোগ করে সিলেবাস সংগ্রহ করতে হবে । এবং সিলেবাসে দেওয়ার নির্দেশনা অনুযায়ী এসাইনমেন্ট তৈরি করতে হবে ।

কলেজ, এইচএসসি, ক্লাস ১১ ও ১২ এর এসাইনমেন্ট কিভাবে করতে হবে?

যেহেতু এই এসাইনমেন্ট টি কলেজের ছাত্র-ছাত্রীদের জন্য সেতু এই এসাইনমেন্ট এর মান স্কুলের ছাত্র-ছাত্রীদের চেয়ে একটু উন্নত করতে হবে। এসাইনমেন্ট এর কভার পেজ সুন্দর হতে হবে এবং রঙিন প্রিন্ট হলে খুব ভালো হয়। অ্যাসাইনমেন্ট হাতে লিখতে হবে নাকি কম্পিউটারে লিখলে চলবে তা কলেজ কর্তৃক নির্ধারণ করে দেওয়া হবে। খেয়াল রাখতে হবে যেন অ্যাসাইনমেন্টের পাতাগুলো স্বচ্ছ এবং পরিষ্কার থাকে সে ক্ষেত্রে অফসেট প্রেস ব্যবহার করা ভালো । কলেজ অ্যাসাইনমেন্ট গুলো একটু ভিন্নধর্মী হাওয়াই প্রশ্নগুলোকে খুব ভালোভাবে অনুধাবন করতে হবে, বুঝতে হবে আসলে কি চাচ্ছে তোমার কাছে । কোন বিষয়ে বক্তব্য প্রদানের ক্ষেত্রে স্পষ্ট ব্যাখ্যা প্রদান করতে হবে । অহেতুক কোনকিছু বা প্রশ্নে যা চাওয়া হয়নি এমন কিছু আলোচনা করা যাবে না । college ,hsc, intermediate, class 11 , class 12 এর এসাইনমেন্ট অন্যদের এসাইনমেন্ট এর চেয়ে অধিক গুরুত্ব বহন করে সুতরাং এসাইনমেন্ট তৈরীর সময় সেদিকে খেয়াল রাখতে হবে । অ্যাসাইনমেন্ট লেখার ভাষা সুন্দর প্রাঞ্জল হতে হবে । আলোচ্য বিষয়ের ওপর নিজের মতামত দেওয়া যেতে পারে এছাড়াও কোন অভিজ্ঞতা থাকলে তা বর্ণনা করা ভালো ।

কলেজ, এইচএসসি, ক্লাস ১১ ও ১২ এর এসাইনমেন্ট কিভাবে সাবমিট করবে?

 এসাইনমেন্ট টি সম্পূর্ণ তৈরি হবার পর এটি যদি হাতের লেখা এসাইনমেন্ট হয় তাহলে তা সরাসরি কলেজ কর্তৃপক্ষের নিকট অথবা শিক্ষকের নিকট হস্তান্তর করতে হবে । আর যদি অ্যাসাইনমেন্ট ইকোনো ডিজিটাল ফরমেটে অ্যাসাইনমেন্ট হয়ে থাকে তবে সেটি ইমেইলের মাধ্যমে কলেজ কর্তৃপক্ষের ইমেইলে বা শিক্ষকের ইমেইল পাঠানো যেতে পারে ।

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MTH 419--Introduction to Abstract Algebra--Homework Assignments

Note: problems in brackets [ ] are to be done, but not turned in.

• Exam I: Monday, Oct 4 completed
• Exam II: Monday, Nov 1 completed
• Exam III: Monday, Nov 29 completed
• Final Exam: Monday, December 13, 11:45AM - 2:45 PM in Math 150
• Contemporary Abstract Algebra, 9th edition, by Joseph Gallian

Assignment XIII--Week of November 29:

• Quiz D1 Wed recitation: No Quiz in recitation this week, due in Gradescope
• Format for quizzes: one definition/statement, one problem/proof
• Quiz D2 Fri recitation: No Quiz in recitation this week, due in Gradescope
• Turn in Problem Set 12 (from the text book), due  Monday 12/06/2021, in Gradescope by 11:59 PM
• Chapter 10: [5], 6, 7, [8], 9, [14], 16, 19, [35]
• Note: some problems ask if there is a homomorphism from G onto H, rather than from G to H. This means that the homomorphism is supposed to be onto .
• homework should be typed, preferably in LaTeX.
• Note: problems listed for an assignment are  to be uploaded to Gradescope by the due date

Assignment XIV--Week of December 6:

• Quiz D1 Wed recitation: Quiz XI  in recitation this week, due in Gradescope
• Covers: Chapter 9, through Example 17; Chapter 10
• Quiz D2 Fri recitation: Quiz XI in recitation this week, due in Gradescope
• Covers: Chapter 9, through Example 17; Chapter 10, beginning of Chapter 11 (depending on where we get on Wednesday)
• Turn in Problem Set 13 (from the text book), due  ->Saturday<- 12/11/2021, in Gradescope by 11:59 PM
• Chapter 10: 40 - 42, 49, 58
• Chapter 11: 2, [6], 12, 21

Final Exam--Monday, December 13

• When 11:45 - 2:45 PM in Math 150
• Covers : 
• Chapters 0 (through page 7 and related homework)
• Chapters 1 - 11
• Chapter 8: skip Examples 4 & 5, skip Theorem 8.3 to end of chapter 8
• Chapter 9: only through Example 16)
• Chapter 11: up to, but not including Lemma 1
• Format: 2 or 3 statements of theorems/definitions; 6 to 12 problems/proofs
• proofs from the book or lecture, or pieces of proofs, may well be on the exam
• homework problems may well be on the exam
• properties left as exercises in class may well be on the exam
• Seating will be assigned
• Exam will be handed out in class
• Must be written in dark ink--black or blue; cross out anything you do not want graded
• Instructor will upload the exams to Gradescope
• Calculators: Only non-programmable, non-graphing, non-alphanumeric are allowed on exams
• Notes: None

Coming Attractions (tentative list of upcoming assignments): May be changed

Assignment i--week of august 30:.

• No classes: Monday, Sept 6 (Labor Day)
• Quiz D1 recitation: Quiz I in recitation, due in Gradescope
• Covers: Division algorithm
• Quiz D2 recitation: Quiz I in recitation, due in Gradescope
• Covers: Division algorithm, Bezout's Lemma (GCD is a linear combination), Euclid's Lemma
• Turn in Problem Set 1: (from the text book), due Tuesday, 09/07/2021, in Gradescope by 11:59 PM
• Chapter 0: 4 - 9, 11- 13, 18 - 21, 30

Assignment II--Week of September 6:

• NO LECTURE ON WEDNESDAY, Sept 8
• Lecture is recorded: UBlearns>Zoom -> cloud recordings sign in with sso. Two recordings from 09/05/21. Start with the older recording.
• Quiz D1 Wed recitation: Quiz II in recitation, due in Gradescope
• Covers: Bezout's Lemma (GCD is a linear combination), Euclid's Lemma, Modular arithmetic
• Quiz D2 Fri recitation: Quiz II in recitation, due in Gradescope
• Covers: Modular arithmetic, symmetries of a square
• Turn in Problem Set 2: (from the text book), due Monday, 09/13/2021, in Gradescope by 11:59 PM
• Chapter 1: [1], 2, 5, [6], 8, 10, 13, 15, 18 Problems in parentheses are to be done but not turned in

Assignment III--Week of September 13:

• Quiz D1 Wed recitation: Quiz III in recitation, due in Gradescope
• Covers: symmetries of square, groups
• Quiz D2 Fri recitation: Quiz III in recitation, due in Gradescope
• Covers: Groups, including U(n)
• Turn in Problem Set 3: (from the text book), due Monday, 09/20/2021, in Gradescope by 11:59 PM
• Chapter 2: 4 - 6, 9 - 12, 17, 22, 25, 26, [31], 32, 37
• Note: Table 2.1 contains a typo. The inverse in C* should have a^2 + b^2 in the denominator for both terms.

Assignment IV--Week of September 20:

• NO LECTURE ON WEDNESDAY, Sept 22
• Lecture is recorded: UBlearns>Zoom -> cloud recordings sign in with sso.
• Quiz D1 Wed recitation: Quiz IV in recitation, due in Gradescope
• Covers: Groups, including U(n), subgroup tests
• Quiz D2 Fri recitation: Quiz IV in recitation, due in Gradescope
• Covers: subgroup tests, cyclic groups
• Turn in Problem Set 4: (from the text book), due Monday, 09/27/2021, in Gradescope by 11:59 PM
• Chapter 3: [1 - 5], 13, 17, 19, 23, 31, [34], 44, 47, 70, 71, 73
• Chapter 4: 2, 3, [4], 5, 7

Assignment V--Week of September 27:

• NO LECTURE ON WEDNESDAY, Sept 29
• Quiz D1 Wed recitation: Quiz V in recitation, due in Gradescope
• Covers: order of elements, cyclic groups
• Quiz D2 Fri recitation: No quiz this week
• Turn in Problem Set 5: (from the text book), due -->Wednesday<-- 10/06/2021, in Gradescope by 11:59 PM
• Chapter 4: [8], 9, [12], 14, [23], 24, 31, 35, 50, 60, 77

Assignment VI--Week of October 4:

• Quiz D1 Wed recitation: No quiz in recitation this week, due in Gradescope
• Quiz D2 Fri recitation: Quiz V in recitation, due in Gradescope
• Covers: Cyclic subgroups, generators
• Turn in Problem Set 6: (from the text book), due Monday 10/11/2021, in Gradescope by 11:59 PM
• Chapter 4: 13, 29, 53, 68, [Try 85]
• Chapter 4: 35, 70

Assignment VII--Week of October 11:

• Quiz D1 Wed recitation: Quiz VI in recitation this week, due in Gradescope
• Covers: Cyclic subgroups, generators, subgroup lattice
• Quiz D2 Fri recitation: Quiz VI in recitation, due in Gradescope
• Covers: subgroup lattice, beginning of permutations
• Turn in Problem Set 7: (from the text book), due ->>Tuesday<<- 10/19/2021, in Gradescope by 11:59 PM
• Chapter 5: 1 - 3, [4 - 8], 9, 13, [15], 16, 17, [19], 23, [26], 28, 29, [32]

Assignment VIII--Week of October 18:

• Quiz D1 Wed recitation: Quiz VII in recitation this week, due in Gradescope
• Covers: Permutations, S_n, A_n
• order of permutations, even/odd permutations, writing a permutation as as a product of disjoint cylces
• writing a permutation as a product of transpositions
• Quiz D2 Fri recitation: Quiz VII in recitation, due in Gradescope
• Permutations, S_n, A_n
• isomorphisms and isomorphic groups
• Turn in Problem Set 8: (from the text book), due Monday 10/25/2021, in Gradescope by 11:59 PM
• Chapter 5: 35, 36, 53, 57, 61, 65
• Chapter 6: [1], [4, 5], 6, 7, 12, [22], 36, 39, 41

Assignment IX--Week of October 25:

• Quiz D1 Wed recitation: Quiz VIII in recitation this week, due in Gradescope
• properties of isomorphisms
• Quiz D2 Fri recitation: no quiz this week
• Turn in Problem Set 8: (from the text book), due -->Tuesday<-- 11/02/2021, in Gradescope by 11:59 PM
• Chapter 6: 51, 53, [try 67]
• Chapter 6: 2, 9, 14, 17 -19, 26, 60

Assignment X--Week of November 1:

• Quiz D1 Wed recitation: no quiz this week
• Quiz D2 Fri recitation: Quiz VIII in recitation this week, due in Gradescope
• Covers: automorphisms
• including Inn(G), Aut(G), Aut(Z), Aut(Z_n)
• Turn in Problem Set 9: (from the text book), due  Monday 11/08/2021, in Gradescope by 11:59 PM
• Chapter 7: 1, [2], 3 - 8, 12, 13, [15, 16], 17 - 19, 22, 26
• Problem 8 has a misprint. Should be: HK = {hk |  h in H, k in K}.

Assignment XI--Week of November 8:

• Quiz D1 Wed recitation: Quiz IX in recitation this week, due in Gradescope
• Covers: automorphisms and cosets
• Quiz D2 Fri recitation: Quiz IX in recitation this week, due in Gradescope
• Covers: cosets
• including normalizer-stabilizer theorem
• Turn in Problem Set 10 (from the text book), due  ->Tuesday<- 11/16/2021, in Gradescope by 11:59 PM
• Chapter 7: 30, 33 , [45], 53, [58], 59
• Chapter 8: 1, 5 - 7, 14, 16, 27. Note: 11, 34 have been deleted from the assignment.

Assignment XII--Week of November 15:

• Quiz D1 Wed recitation: Quiz X in recitation this week, due in Gradescope
• Covers: cosets, direct product of groups
• including orbit-stabilizer theorem
• Quiz D2 Fri recitation: Quiz X in recitation this week, due in Gradescope
• Covers: direct product of groups, normal subgroups
• Turn in Problem Set 11 (from the text book), due  Monday 11/22/2021, in Gradescope by 11:59 PM
• Chapter 9: 2, 5, 6, 9, [10 - 12], 17, [18, 19], 22, 37, [43], 51, 59, 61, 67, [71]

Exam I--Monday, October 4

• When 11:30 - 12:20 PM in Math 150
• Covers :  Chapters 0 (through page 7 and related homework), 1 - 4 (through Theorem 4.1 only)
• 1 or 2 statements/definitions; 3 or 4 problems/proofs

Exam II--Monday, November 1

• Covers :  Chapters 4 - 6 (up to, but not including Automorphisms on  page 128)
• be sure to review center, centralizer, order of a group

Exam III--Monday, November 29

• Covers :  Chapters 6 - 9 (Chapter 8: skip Examples 4 & 5, skip Theorem 8.3 to end of chapter 8; Chapter 9: only through Example 16)

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The Independent Nuclear News Agency

Fast reactors / china and russia sign agreement to continue cooperation.

By David Dalton 23 March 2023

According to the International Atomic Energy Agency, global interest in fast reactors has been growing since their inception in 1960 because they can provide “efficient, safe and sustainable energy”. Their closed fuel cycle – a fuel cycle that reuses spent fuel – can support long-term nuclear power development as part of the world’s future energy mix and decrease the burden of nuclear waste.

Russia and China are cooperating on several reactor projects, including construction of Russian VVER plants at Tianwan and Xudabao in China and the supply of fuel for China’s experimental fast neutron reactor, the CFR-600 in the southeastern province of Fujian.

Russia’s MBIR multipurpose fast neutron research reactor is under construction at the Research Institute of Atomic Reactors site at Dimitrovgrad in the Ulyanovsk region of western Russia and is scheduled for completion in 2028.

In 2021, Russia’s president Vladimir Putin said Moscow was considering the possibility of China joining a consortium for construction of the plant.

In October, Russia’s state nuclear fuel company Tvel said it had begun deliveries of nuclear fuel for China’s CFR-600. In 2021, Tvel started up a production facility to fabricate fuel for the CFR-600. The facility is part of the Elemash Machine-Building Plant, a Tvel plant in Elektrostal, near Moscow.

Fuel manufactured by Russia for China’s CFR-600 fast neutron Reactor. Courtesy Tvel.

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Jointly Building a Community of Shared Future for All Humankind

By staff reporter AN XINZHU

THE world is facing serious dilemmas in its push for further growth, such as insufficient impetus to drive economic growth, the gap between rich and poor, prevalence of protectionism, and global governance that needs to be improved. The questions remains: How to work together to promote common prosperity and inject new impetus to boost sustained economic growth?

“In promoting mutually beneficial cooperation, we should raise awareness on building a community of shared future for all humankind. A country should accommodate the legitimate concerns of others when pursuing its own interests; and it should promote common development of all countries when advancing its own. Countries should establish a new type of global development partnership that is more equitable and balanced, stick together in times of difficulty, share all rights as well as shoulder all obligations, and boost the common interests of humankind.” This is quoted from the report adopted by the 18th National Congress of the Communist Party of China (CPC) in November 2012, which attracted the world’s attention.

An Increasingly Clear Notion

“It is a world where countries are linked with and dependent on one another at a level never seen before. Humankind, by living in the same global village within the same time and space where history and reality meet, has increasingly emerged as an intimate community of shared future,” said Chinese President Xi Jinping. He first made the remarks about sharing with the world China’s perspective on the future development of human civilization in his speech delivered at Moscow State Institute of International Relations during his first overseas trip after becoming China’s president in March 2013.

The response from international society showed that Xi’s understanding of the future development of the whole world and human society is “far-sighted.” The concept of building a community of shared future for all humankind has struck a chord with the rest of the world.

In a series of conferences and major events in the ensuing years, Xi Jinping has expounded on the notion repeatedly. All the remarks he has made and what he has called for over the past five years, including “an intimate community where everyone is interlinked,” “there is only one Earth in the universe and we, humankind, have only one homeland,” “foster a keen sense of a global community of shared future,” “let the awareness of a community of common destiny take root in the neighboring countries,” “create and fulfill an Asia-Pacific dream,” and “move towards a community of common destiny and a new future for Asia,” have made the concept of “building a community of shared future for all humankind” a global notion that has surpassed national boundaries and the constraints of nationality and ideology. China’s influence on regional cooperation and global governance has been expanded from merely an institutional aspect to a spiritual level.

On May 18, 2015, People’s Daily, the flagship newspaper of the CPC, published an article by Guo Jiping, to elaborate on the concept of “a community of shared future for all humankind.” The article pointed out that China has expressed its wish for peaceful development through this global notion that has surpassed the constraints of nationality and ideology.

The article, entitled “Promise of a Better Future for the World,” noted that there are divergences and conflicts when pursuing different state interests, religious beliefs, ideologies, and social systems. However, we are all humans, and people are the subject of any historical activity. Countries with different beliefs, systems, and ethnic groups could coexist peacefully and compete with each other equally and in an orderly fashion, so that the common interests outweigh divergences and confrontations and the people can rationally decide the future of the world.

A New Model of International Relations Based on Win-win Cooperation

The famous British historian Arnold Joseph Toynbee said: “If the CPC can blaze a new path in its social and economic strategies, then it can prove its capacity to endow a gift on China and the world. The gift will be a combination of the vitality of modern Western society and the stability of traditional Chinese culture.”

The notion of “building a community of shared future for all humankind” is probably part of this gift.

In September 2015, Xi Jinping gave a comprehensive elaboration on the notion when he attended summits marking the 70th anniversary of the United Nations’ founding in New York. He pointed out that efforts should be made to establish a partnership in which countries treat each other as equals with mutual consultations and mutual understanding, build a security pattern featuring fairness and justice that is contributed and shared jointly by all nations, strive for open, innovative, and inclusive development that benefits all, advance inter-civilization exchanges featuring inclusiveness and harmony amid diversity, and build an ecosystem that puts nature and green development first.

His elaboration has made the overall plan for building a community of shared future for all humankind clear, depicted the bright prospects of international relations, and also represented the innovation of major country diplomacy with Chinese characteristics.

Following the traditional Chinese doctrine of “the unity of knowing and acting,” China has become a pioneer and contributor in realizing this notion through its actions. Over the past 30-odd years since the implementation of the reform and opening-up policy, China has climbed from the 10th to second largest economy in the world, with its contribution to world economic growth remaining at around 30 percent annually, proving its commitment to development. Meanwhile, China has lifted more than 700 million people out of poverty in the past three decades — which amounts to more than 70 percent of global poverty reduction over that period.

From a political aspect, China insists on a new approach to state-to-state relations that features dialogue rather than confrontation and partnerships instead of alliances. Efforts should be made to seek common ground while shelving differences. Countries need to respect other countries’ unique social systems and development paths, as well as respect each other’s core interests and major concerns. Countries may differ in size, strength, or level of development, but we should discuss and look for solutions together on matters that involve us all. China takes the lead among major countries to make “establishing a partnership” a guiding principle of building international relations. So far, it has established partnerships with over 80 countries, regions, and regional organizations. A partnership network between China and the rest of the world has begun to take shape.

On security issues, China calls for abandoning the outdated Cold War mentality in all forms, and fostering a new vision of common, comprehensive, cooperative, and sustainable security. As people of all countries become increasingly interdependent, no country could ensure its own security without also ensuring the security of other countries or of the wider world. Therefore, all countries should play a constructive role in safeguarding global and regional stability. Guided by the new vision of security, China is working hard to provide solutions to regional hotspot issues. It has actively participated in the UN’s peacekeeping operations by sending the highest number of peacekeepers among all five permanent members of the Security Council and contributing the greatest amount of funds to the UN peacekeeping budget as a developing country.

Regarding the economy, China calls for an open, innovative, and inclusive development that benefits all, so to achieve win-win cooperation and common development. China supports the establishment of a more open economy and firmly boosts free trade and investment by promoting the facilitation of trade and investment and opposing protectionism. Against the anti-globalization movement, China calls for eliminating gaps among countries and allowing the people around the world to share development fruits together.

For cultural exchanges, China strives to promote exchanges among different civilizations and different development models, so that we can learn from each other’s best potential for common progress. “Different civilizations represent different wisdoms and are made up of the contributions from different nationalities. They are equal, none of them superior or inferior. We should respect all civilizations, treat each other as equals, learn from each other, and take in each other’s excellent essence so as to promote human civilization towards creative development,” said Xi Jinping in a speech delivered at the Peruvian Congress in November 2016.

To encourage green development, China has made significant efforts to promote the implementation of the United Nations Framework Convention on Climate Change (UNFCCC). China takes the lead among the world’s biggest carbon emitters in starting domestic procedures to ratify the Paris Agreement. Patricia Espinosa Cantellano, executive secretary of the UNFCCC, said during her visit to China at the end of August 2017 that China is one of the most important leaders in tackling global climate change, noting that the country has made positive efforts in line with its green development concept. She added that China plays a leading role, especially in promoting the implementation of the Paris Agreement.

Sincere Partner for Developing Countries

Building a community of shared future for all humankind starts from surrounding countries. The Boao Forum for Asia witnessed the concept of an Asian community of shared future.

During the annual meeting of the Boao Forum for Asia in 2013, Chinese President Xi Jinping stressed “firmly establishing the consciousness of a community of shared future.” President Xi then delivered an important speech in the Indonesian parliament, and proposed five major initiatives to build the “China-ASEAN community of shared future.” At the Central Conference on Work Relating to Foreign Affairs in 2014, President Xi stressed the need to create a “community of shared future in surrounding countries.”

The theme of the Boao Forum for Asia in 2015 was “Asia’s New Future: Towards a Community of Shared Future.” At the opening ceremony, President Xi once again explained the proposition to “march towards the community of shared future, and create a new future in Asia.” The forum is regarded as the symbol of Asian solidarity by foreign media.

The concept also reaches Africa and Latin America across the ocean.

In March 2013, when President Xi gave an address in Tanzania, he said that China and Africa have the same historical experience, development tasks and strategic interests of a community of shared future. And his 30-minute speech won 30 rounds of applause.

In July 2014, Xi Jinping visited Brazil and attended the meeting with leaders of Latin American and Caribbean countries. In his speech entitled “Strive to Build a Community of Shared Future,” he proposed a new pattern between China and Latin America, and called for the creation of the “Sino-Latin American community of shared future.”

As a developing country, China’s diplomatic work has never neglected the common progress of other developing countries, believing that only joint self-improvement can allow developing countries to go to the center of the world stage. During the 2016 G20 Summit in Hangzhou, China extensively invited developing counties to participate in and to enhance the representation and speaking right of the developing countries at the G20. During the BRICS Leaders Meeting in 2017, the Chinese side held a dialogue between emerging market economies and developing countries, with the aim of building a network of partners and building a community of common development and shared future.

The concept of “community of shared future” has extended from the “real world” to “virtual space.”

According to international media, to build a community of shared future has become China’s new diplomatic strategy, even the new blueprint to establish a fair international order.

Contributing Chinese Wisdom

“The community of shared future” is rooted in the inheritance of thousands of years of Chinese culture.

Chinese Foreign Minister Wang Yi pointed out that the concept is rooted in the long history of Chinese civilization and China’s magnificent diplomatic practice, in line with various countries’ sincere desire and lofty pursuit of peace, development and cooperation.

The Chinese people have said, “The world is a commonwealth.” The community of shared future is not only the resonance of the Chinese philosophy of life, diplomatic traditions and the contemporary era, but also the pursuit of better life of peoples of various countries.

In the China Hall of the United Nations Headquarters hang two Chinese paintings – the Interactive World and the Home of Us All. According to former United Nations Secretary-General Ban Ki-moon, the two paintings reflect the important role of China in the peaceful development of the world. China is not only the advocator of community of shared future, but also the responsible practitioner.

On September 28, 2015, in the UN headquarters, President Xi stated that “we are going to inherit and carry forward the purposes and principles of the Charter of the United Nations, build a new type of international relations centered on win-win cooperation, and build a community of shared future for mankind.” This was the first time that Xi Jinping had elaborated the concept at the United Nations.

On January 18, 2017, President Xi delivered an important speech entitled “Jointly Build a Community of Shared Future for Mankind” in the United Nations Office at Geneva, and clearly put forward China’s approach of “building a community of shared future to achieve win-win results.” In February, the concept was first written into the United Nations resolution. One month later, at the 34th meeting of United Nations Human Rights Council, China, on behalf of 140 countries, issued a joint statement entitled “Work Together to Promote and Protect Human Rights and Build a Community of Shared Future for Humankind.”

China has not only provided a unique vision to promote the progress of world civilization, but also put forward the “China approach” to this idea.

Proposed more than three years ago, the Belt and Road Initiative has entered the new stage of action and implementation. The circle of friends along the Belt and Road is expanding. Since establishment, the Asian Infrastructure Investment Bank has gained plentiful and substantial achievements. As a growing number of Chinese initiatives have become international consensus, a series of constructive public goods have benefited the world.

UN Secretary-General Antonio Guterres highly appreciates the idea put forward by President Xi. “China has become an important pillar of multilateralism, and the purpose of multilateralism is to establish a community of shared future for all humankind ,” he said.

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MLB Fans Call for Angel Hernandez’s Job After Another Truly Horrific Game

• Author: Tim Capurso

In this story:

MLB umpire Angel Hernandez, notorious for his wishy-washy strike zone, was in rare form during Friday night's game between the Texas Rangers and Houston Astros, a 12-8 victory for the visiting Rangers.

Hernandez's performance definitely didn't pass the eye test, as he left the Rangers broadcasters practically gobsmacked after calling three consecutive strikes on pitches that appeared to be outside the zone.

The veteran umpire didn't fare much better on a deeper dive into the numbers behind his performance Friday, as Umpire Scorecards, an account on X (formerly Twitter) that grades the efforts of the game's officials, rated his performance quite harshly.

Umpire: Angel Hernandez Final: Rangers 12, Astros 8 #StraightUpTX // #Relentless #TEXvsHOU // #HOUvsTEX More stats for this game 👇 https://t.co/RJKSQogI6y pic.twitter.com/uRM1ykGr3R — Umpire Scorecards (@UmpScorecards) April 13, 2024

MLB fans proceeded to react to Hernandez's less-than-stellar performance, which included 12 called strikes that Umpire Scorecard regarded as true balls, exactly as one would expect.

WHY IS HE NOT FIRED ALREADY — . (@HalToHell) April 13, 2024

Yet another flawless performance by Angel Hernandez — Sill E. Gewce (@ParahoyitsAJ) April 13, 2024

I’m really not sure why or how he’s still allowed to umpire MLB games. — Texas Rangers Insider (@RangersInsiders) April 13, 2024

78% accuracy — Space City Sports (@LiveSCS) April 13, 2024

Hernandez has been an MLB umpire since 1991. It's safe to say that the majority of baseball fans are ready for his tenure to end.

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Wednesday Briefing

Israel’s difficult choices on Iran.

By Daniel E. Slotnik

For Israel, every response to Iran carries risks

Israeli leaders were weighing a set of possible responses to Iran’s airstrike, aimed at achieving different outcomes: deterring a future attack, placating Israel’s U.S. allies and avoiding all-out war.

Iran’s attack on Israel, involving hundreds of ballistic missiles and exploding drones, changed the unspoken rules in the archrivals’ long-running shadow war. Until last weekend, neither country had directly carried out a major strike on the other’s territory.

Israel’s fractious war cabinet is considering options like hitting an Iranian target in a third country (similar to the strike on Iranian military officers in Syria that prompted Iran’s barrage), attacking a mostly symbolic target inside Iran, and carrying out a cyberattack on Iranian infrastructure. All of those choices have potential drawbacks.

After three days of meetings, the cabinet has yet to decide on a response. It is expected to reconvene today.

The U.S. announced that it was trying to find ways to cut off Iran’s access to military components and to further restrict its oil exports, while E.U. ministers discussed expanding sanctions on Iran’s weapons program.

Seven jurors were chosen for Trump’s Manhattan trial

The daunting work of selecting a jury for the first criminal trial of a former U.S. president gained momentum yesterday, as seven New Yorkers were picked to sit in judgment of Donald Trump.

The lawyers quizzed prospective jurors on their ability to remain impartial about a defendant who may be the most polarizing figure in U.S. politics.

The wrangling underscored the challenge of picking a jury in a city where Trump, the presumptive Republican nominee for president, is deeply unpopular. If jury selection continues at this pace, opening arguments will probably begin on Monday, the judge overseeing the case said.

For more, see how Melania Trump is viewing the trial , read Day 2 takeaways and the 42 questions lawyers asked possible jurors .

Separately, Trump has held several private meetings with foreign leaders, including many who share his brand of politics.

Vulnerabilities create a dire situation for Ukraine

The three most critical challenges for Ukraine have been evident for months: a lack of ammunition, a shortage of well-trained troops and dwindling air defenses.

Now, as the Kremlin intensifies its assaults and Ukrainian intelligence suggests a major spring or summer Russian offensive, each of those problems is heightening the risk that Moscow’s forces will push through Ukraine’s defenses.

Aid: The U.S. House speaker encountered stiff resistance from Republicans as he embarked on a complicated strategy to push aid to Ukraine and Israel through Congress.

China’s economy picked up momentum

China’s economy grew 1.6 percent in the first quarter of 2024, after the country built more factories and exported huge amounts of goods to counter a severe real estate crisis and sluggish spending at home.

But many foreign countries fear that a flood of Chinese shipments could undermine their own manufacturing industries and lead to layoffs.

“China may have found a way to blunt the effects of its housing market crisis,” my colleague Keith Bradsher told us, “but only if Beijing can persuade other countries to buy more manufactured goods than ever from China.”

MORE TOP NEWS

Germany: In meetings with Chinese leaders, including Xi Jinping, Chancellor Olaf Scholz raised concerns about Beijing’s surge of exports to Europe and its support for Russia.

U.N.: Members of a commission said that Israel was obstructing their efforts to investigate possible human rights violations by Hamas and Israel during the Oct. 7 attack and the ensuing Gaza war.

Brussels: The city’s mayor tried to break up a gathering of “anti-woke” conservatives from across Europe, amplifying their argument that cancel culture was running amok .

U.S.: The Supreme Court seemed wary of letting prosecutors use a federal obstruction law to charge hundreds of rioters involved in the Capitol attack on Jan. 6, 2021.

Denmark: The old stock exchange building in downtown Copenhagen partly collapsed in a large fire.

Olympics: The Paris Olympics organizers are preparing for cyberattacks by hosting war games and paying “bug bounties” to hackers.

Beijing Half Marathon: Organizers were investigating after runners from Kenya and Ethiopia appeared to let a Chinese competitor pass them just before the finish line.

Economy and Technology

A.I.: Microsoft said it would invest $1.5 billion in G42, an Emirati company with ties to China, as Washington and Beijing vied for tech influence in the Persian Gulf.

I.M.F.: The International Monetary Fund projected that global output would hold steady at 3.2 percent in 2024 but warned of risks like stubborn inflation and rising protectionism.

MORNING READ

The word “polycule” is a synthesis of polyamory — engaging in multiple romantic relationships — and molecule, to suggest an intricate structure formed of people with overlapping deep attachments: romantic, sexual, sensual, platonic.

It’s difficult to describe a polycule, which “family” and “network” don’t quite capture. Here’s how people in a polycule in Boston think of their arrangement .

CONVERSATION STARTERS

“Cats are just catastrophic”: Feral felines take a heavy toll on wildlife in Australia.

Venice Biennale: Israel’s representative said she wouldn’t open her show until Israel and Hamas agreed to a cease-fire and hostage release.

Frida Uncensored: A mother-and-baby-care brand is working with Asa Akira, a porn actress, to make educational videos about its products .

SPORTS NEWS

FIFA rule changes: Why European soccer matches might finally be coming to the U.S .

Rebuilding Old Trafford: What the local residents and businesses think.

Fitness and pride: Rafael Nadal’s clay-court swan song .

ARTS AND IDEAS

Taylor swift and beyoncé avoided a chart collision.

Beyoncé and Taylor Swift, the 21st century’s pre-eminent pop stars, have often been cast as competitors if not rivals, a story line partly rooted in misogyny and amplified by dueling armies of superfans.

For their part, the two pop queens have regularly dispelled that notion, including by not challenging each other on the charts. This year, the release dates for their new albums, Beyoncé’s “Cowboy Carter” and Swift’s “The Tortured Poets Department,” are weeks apart, giving each of them the space to dominate the charts .

RECOMMENDATIONS

Cook: Arroz con pollo can be simple or not. It depends on your ingredients.

Listen: These charming audiobooks are ideal bedtime stories for adults.

Exercise: This is how you can turn your bike ride into a workout.

Plan: Which dinner party menu sounds good to you: simple or elaborate?

Play the Spelling Bee . And here are today’s Mini Crossword and Wordle . You can find all our puzzles here .

That’s it for today’s briefing. Thank you for spending part of your morning with us, and see you tomorrow. — Dan

P.S. Lauren Jackson is the new associate editor of The Morning newsletter.

You can reach Dan and the team at [email protected] .

Daniel E. Slotnik is a general assignment reporter on the Metro desk and a 2020 New York Times reporting fellow. More about Daniel E. Slotnik