Math 230 D
Homework 5
Fall 2012
Drew Armstrong
Problem 1. Recall that a b mod n means that n|(a b). Use induction to prove that for
all n 2, the following holds:
if a1 , a2 , . . . , an Z such that each ai 1 mod 4, then a1 a2 an 1 mod 4.
[Hint: Call the s
Math 230 D
Homework 3
Fall 2012
Drew Armstrong
Problem 1. Let X and Y be nite sets.
(a) If there exists a surjective function f : X Y , prove that |X| |Y |.
(b) If there exists an injective function g : X Y , prove that |X| |Y |.
(c) If there exists a bij
Math 230 D
Homework 5
Fall 2012
Drew Armstrong
Problem 1. Recall that a b mod n means that n|(a b). Use induction to prove that for
all n 2, the following holds:
if a1 , a2 , . . . , an Z such that each ai 1 mod 4, then a1 a2 an 1 mod 4.
[Hint: Call the s
Math 230 D
Homework 6
Fall 2012
Drew Armstrong
Problem 1. For each integer n 0, let P (n) be the statement: any set of size n has 2n
subsets. Use induction to prove that P (n) is true for all n 0. [Hint: Let A be an arbitrary
set of size n and let x A be
Math 230 E
Homework 4
Fall 2013
Drew Armstrong
Problem 1. Prove that for all integers a, b Z we have
(ab = 0)
=
(a = 0 or b = 0).
You may assume the following axioms: (1) For all x, y, z Z, if x < y and z > 0 then xz < yz.
(2) For all x, y, z Z, if x < y
Math 230 E
Homework 6
Fall 2013
Drew Armstrong
Problem 1 (Binomial Theorem). We proved in class that for all n 0 we have
n
n
(1 + x) =
k=0
n!
xk .
k! (n k)!
Use this to prove that for all integers a, b Z we have
n
n
(a + b) =
k=0
n!
ank bk .
k! (n k)!
[Hi
Math 230 E
Homework 5
Fall 2013
Drew Armstrong
Problem 1. Use induction to prove that for all integers n 1 we have
13 + 23 + 33 + + n3 = (1 + 2 + + n)2 .
This result appears in the Aryabhatiya of Aryabhata (499 CE, when he was 23 years old).
[Hint: You ma
Math 230 E
Homework 3
Fall 2013
Drew Armstrong
Problem 1. Let X and Y be nite sets.
(a) If there exists a surjective function f : X Y , prove that |X| |Y |.
(b) If there exists an injective function g : X Y , prove that |X| |Y |.
(c) If there exists a bij
Math 230 E
Homework 2
Fall 2013
Drew Armstrong
Problem 1. Practice with the axioms of Z. For the following exercises I want you to
give Euclidean style proofs using the axioms of Z from the handout. That is, dont assume
anything and justify every tiny lit
Math 230 D
Homework 1
Fall 2012
Drew Armstrong
Problem 1. In the Lilavati, the Indian mathematician Bhaskara (11141185) gave a one-word
proof of the Pythagorean theorem. He said: Behold!
Add words to the proof. Your goal is to persuade a high school stude
Math 230 D
Homework 4
Fall 2012
Drew Armstrong
Problem 1. Let a, b, c Z be integers. Prove the following.
(a) If a|b and b|c, then a|c.
(b) If a|b and a|c, then a|(bx + cy) for any x, y Z.
(c) If a|b and b|a, then a = b.
Problem 2. Given a, b Z not both z
Math 230 D
Homework 2
Fall 2012
Drew Armstrong
Problem 1.
(a) Prove that a product of odd numbers is odd.
(b) Prove that 3n is odd for all integers n 1. [Its easy to prove that, say, 3101 is odd. But
how will you prove it for innitely many dierent n witho
Math 230 E
Exam 3 Fri Dec 2
Fall 2011
Drew Armstrong
There are 4 problems, worth 6 points each. There is 1 bonus point for writing your name.
This is a closed book test. Anyone caught cheating will receive a score of zero.
1. [6 points]
(a) How many words
Math 230 D
Homework 6
Fall 2012
Drew Armstrong
Problem 1. For each integer n 0, let P (n) be the statement: any set of size n has 2n
subsets. Use induction to prove that P (n) is true for all n 0. [Hint: Let A be an arbitrary
set of size n and let x A be
Math 230 D
Homework 1
Fall 2012
Drew Armstrong
Problem 1. In the Lilavati, the Indian mathematician Bhaskara (11141185) gave a one-word
proof of the Pythagorean theorem. He said: Behold!
Add words to the proof. Your goal is to persuade a high school stude
Math 230 D
Exam 1 Fri Sept 21
Fall 2012
Drew Armstrong
There are 4 problems, worth 5 points each. This is a closed book test. Anyone caught cheating
will receive a score of zero.
Problem 1. Let P and Q be Boolean variables.
(a) Draw the truth table for th
Math 230 D
Exam 2 Wed Oct 24
Fall 2012
Drew Armstrong
There are 4 problems, worth 5 points each. This is a closed book test. Anyone caught
cheating will receive a score of zero.
Problem 1.
(a) Use the Extended Euclidean Algorithm to compute gcd(12, 7). Co
Math 230 D
Homework 4
Fall 2012
Drew Armstrong
Problem 1. Let a, b, c Z be integers. Prove the following.
(a) If a|b and b|c, then a|c.
(b) If a|b and a|c, then a|(bx + cy) for any x, y Z.
(c) If a|b and b|a, then a = b.
Proof. To prove (a), suppose that
Math 230 D
Homework 3
Fall 2012
Drew Armstrong
Problem 1. Let X and Y be nite sets.
(a) If there exists a surjective function f : X Y , prove that |X| |Y |.
(b) If there exists an injective function g : X Y , prove that |X| |Y |.
(c) If there exists a bij
Math 230 D
Homework 2
Fall 2012
Drew Armstrong
Problem 1.
(a) Prove that the product of two odd numbers is odd.
(b) Prove that 3n is odd for all integers n 1. [Its easy to prove that, say, 3101 is odd. But
how will you prove it for innitely many dierent n
Math 230 E
Homework 1
Fall 2013
Drew Armstrong
Problem 1. Proposition I.5 in Euclid has acquired the name pons asinorum, which translates
as bridge of asses or bridge of fools. Apparently, many students never got past this proposition. (I would say that t