+ Homework for MATH 3163-002, Spring 2020 Homework assignments
(MATH 3163-002, Spring 2020)

ARCHIVE PAGE

This page is not updated any more.

Last update: Tuesday, April 14, 2020

Disclaimer: The information below comes with no warranty. If, due to typographical error, there is a discrepancy between the exercises announced in class and the ones below, or this page is not completely up to date, the required homework consists of those exercises which were announced in class. Check for the time of last update above. If, by my mistake, a wrong exercise shows up below, I will allow you extra time to hand in the exercise that was announced in class. If, however, exercises are missing because this page is not up to date, it is your responsibility to contact me before the due date. (No extra time will be allowed in that case.) This page is up to date if the last update happened after the last class before the next due date.

Notation: In the table below, 1.1/1a means exercise 1, part a, in section 1.1.

No. Date due: Problems:

12 Tue 4/21 5.2/2,8 5.3/8.
Bonus: 5.3/9b (B09, 5 points)

11 Tue 4/14 4.5/1b, 1d, 5.1/4,10,12.

10 Tue 4/7 4.3/22a, 4.4/12, 14a, 19a
Bonus problem: 4.4/24 (B08, 5 points).

9 Tue 3/24 4.2/5b, 8, 9, 12. Note that a hint is given for 4.2/9 in the back of the book, but I would like to see all details.
Bonus problem: Prove the product rule for derivatives for polynomials with coefficients in an arbitrary field. (11.5/5ab, 5 points). (B07, 5 points) (Note: this is B07, the number B10 was a typo!).

8 Tue 3/17 4.1/5d,12,18,20; 4.2/2,10. This assignment and all subsequent assignments have to be submitted through Canvas, online. Please log onto Canvas for details!
Bonus problem:

Let R be a commutative ring with a multiplicative identity element. We say that a is an associate of b in R if there is a unit c satisfying a=bc. Prove that the relation "a is an associate of b" is an equivalence relation. (B06, 5 points)

7 Tue 3/10 3.3/2,8,12ab,24b,26.

6 Tue 2/25 3.1/11bc, 22 (only prove it is a ring); 3.2/3b,20, 22b.
Bonus problems:

Prove that the square root of a positive integer is either an integer or it is irrational. (B02, 5 points)
2.1/20 for n=6 (B03, 5 points)
2.3/13b (B04, 5 points)
2.3/14a (B05, 5 points)

5 Tue 2/18 2.3/8b (use the equations [8][x]=[2], [3][x]=[1] and [6][x]=[4] in ℤ₁₂). 3.1/6b, 10.
Extra credit: Prove by induction that ¹/_{1·
2}+¹/_{2· 3}+ … + ¹/_{(n-1)· n} =1- ¹ /_n holds for n ≥ 1. (10 points, due 2/18)

4 Tue 2/11 2.2/2, 10 (parts 8 and 9), 16a, 16d; 2.3/4a, 4d.

3 Tue 2/4 2.1/4,10,14a,16.
Bonus problem:

(B01, 5 points) 1.2/33.

2 Tue 1/28 1.2/15c (also write the greatest common divisor of 1003, and 456 as 1003 m+ 456 n, see the file notes-0121.pdf on Canvas for help), 1.2/34a; 1.3/6,8b, 14.
Board problem:

Prove that a divides b if and only if a divides -b.

1 Tue 1/21 1.1/8,11.
Board problems:

Prove by induction that 1²+2²+...+n²=n(n+1)(2n+1)/6.
The sequence a₀, a₁, a₂,... is given by the initial conditions a₀=4 , a₁=7 and by the recurrence a_n+2=3 · a_n+1-2 ·a_n.
Prove by induction that a_n=1+3 · 2ⁿ

No.	Date due:	Problems:
12	Tue 4/21	5.2/2,8 5.3/8. Bonus: 5.3/9b (B09, 5 points)
11	Tue 4/14	4.5/1b, 1d, 5.1/4,10,12.
10	Tue 4/7	4.3/22a, 4.4/12, 14a, 19a Bonus problem: 4.4/24 (B08, 5 points).
9	Tue 3/24	4.2/5b, 8, 9, 12. Note that a hint is given for 4.2/9 in the back of the book, but I would like to see all details. Bonus problem: Prove the product rule for derivatives for polynomials with coefficients in an arbitrary field. (11.5/5ab, 5 points). (B07, 5 points) (Note: this is B07, the number B10 was a typo!).
8	Tue 3/17	4.1/5d,12,18,20; 4.2/2,10. This assignment and all subsequent assignments have to be submitted through Canvas, online. Please log onto Canvas for details! Bonus problem: Let R be a commutative ring with a multiplicative identity element. We say that a is an associate of b in R if there is a unit c satisfying a=bc. Prove that the relation "a is an associate of b" is an equivalence relation. (B06, 5 points)
7	Tue 3/10	3.3/2,8,12ab,24b,26.
6	Tue 2/25	3.1/11bc, 22 (only prove it is a ring); 3.2/3b,20, 22b. Bonus problems: Prove that the square root of a positive integer is either an integer or it is irrational. (B02, 5 points) 2.1/20 for n=6 (B03, 5 points) 2.3/13b (B04, 5 points) 2.3/14a (B05, 5 points)
5	Tue 2/18	2.3/8b (use the equations [8][x]=[2], [3][x]=[1] and [6][x]=[4] in ℤ₁₂). 3.1/6b, 10. Extra credit: Prove by induction that ¹/_{1· 2}+¹/_{2· 3}+ … + ¹/_{(n-1)· n} =1- ¹ /_n holds for n ≥ 1. (10 points, due 2/18)
4	Tue 2/11	2.2/2, 10 (parts 8 and 9), 16a, 16d; 2.3/4a, 4d.
3	Tue 2/4	2.1/4,10,14a,16. Bonus problem: (B01, 5 points) 1.2/33.
2	Tue 1/28	1.2/15c (also write the greatest common divisor of 1003, and 456 as 1003 m+ 456 n, see the file notes-0121.pdf on Canvas for help), 1.2/34a; 1.3/6,8b, 14. Board problem: Prove that a divides b if and only if a divides -b.
1	Tue 1/21	1.1/8,11. Board problems: Prove by induction that 1²+2²+...+n²=n(n+1)(2n+1)/6. The sequence a₀, a₁, a₂,... is given by the initial conditions a₀=4 , a₁=7 and by the recurrence a_n+2=3 · a_n+1-2 ·a_n. Prove by induction that a_n=1+3 · 2ⁿ