# MATH 300 Fall 2018 Assignment 14

## Direct Proofs

The following is the list of problems for Chapter 5 of the

*Book of Proof*(page 110). There is a forum open at the end, so you can ask questions. It is a great way to interact with the instructor and with other students in your class, should you need some assistance with any question. Please,**do not post solutions**.A. Use the method of contrapositive proof to prove the following statements. (In each case you should also think about how a direct proof would work. You will find in most cases that contrapositive is easier.)

- Suppose \( n \in \mathbb{Z} \). if \( n^2 \) is even, then \( n \) is even.
- Suppose \( n \in \mathbb{Z} \). if \( n^2 \) is odd, then \( n \) is odd.
- Suppose \( a, b \in \mathbb{Z} \). If \( a^2(b^2-2b) \) is odd, then \( a \) and \( b \) are odd.
- Suppose \( a, b, c \in \mathbb{Z} \). If \( a \) does not divide \( bc \), then \( a \) does not divide \( b \).
- Suppose \( x \in \mathbb{R} \). If \( x^2+5x<0 \), then \( x < 0 \).
- Suppose \( x \in \mathbb{R} \). If \( x^3-x>0 \) then \( x > -1 \).
- Suppose \( a,b \in \mathbb{Z} \). If both \( ab \) and \( a+b \) are even, then both \( a \) and \( b \) are even.
- Suppose \( x \in \mathbb{R} \). If \( x^5-4x^4+3x^3-x^2+3x-4 \geq 0 \), then \( x \geq 0 \).
- Suppose \( n \in \mathbb{Z} \). If \( 3 \!\nmid\! n^2 \), then \( 3 \!\nmid\! n \).
- Suppose \( x, y, z \in \mathbb{Z} \) and \( x \neq 0 \). If \( x \!\nmid\! yz \), then \( x \!\nmid\! y \) and \( x \!\nmid\! z \).
- Suppose \( x, y \in \mathbb{Z} \). If \( x^2(y+3) \) is even, then \( x \) is even or \( y \) is odd.
- Suppose \( a \in \mathbb{Z} \). If \( a^2 \) is not divisible by 4, then \( a \) is odd.
- Suppose \( x \in \mathbb{R} \). If \( x^5 + 7x^3 + 5x \geq x^4 + x^2 + 8 \), then \( x \geq 0 \).

B. Prove the following statements using either direct or contrapositive proof. Sometimes one approach will be much easier than the other.

- If \( a, b \in \mathbb{Z} \) and \( a \) and \( b \) have the same parity, then \( 3a +7 \) and \( 7b-4 \) do not.
- Suppose \( x \in \mathbb{Z} \). If \( x^3-1 \) is even, then \( x \) is odd.
- Suppose \( x \in \mathbb{Z} \). If \( x+y \) is even, then \( x \) and \( y \) have the same parity.
- If \( n \) is odd, then \( 8 \vert (n^2-1) \).