# MATH 300 Fall 2018 Assignment 15

## Direct Proofs

The following is the list of problems for Chapter 6 of the

*Book of Proof*(page 118). 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 proof by contradiction to prove the following statements. (In each case, you should also think about how a direct or contrapositive proof would work. You will find in most cases that proof by contradiction is easier.)

- Suppose \( n \in \mathbb{Z} \). If \( n \) is odd, then \( n^2 \) is odd.
- Suppose \( n \in \mathbb{Z} \). If \( n^2 \) is odd, then \( n \) is odd.
- Prove that \( \sqrt[3]{2} \) is irrational.
- Prove that \( \sqrt{6} \) is irrational.
- Prove that \( \sqrt{3} \) is irrational.
- If \( a, b \in \mathbb{Z} \), then \( a^2-4b-2 \neq 0 \).
- If \( a, b \in \mathbb{Z} \), then \( a^2-4b-3 \neq 0 \).
- Suppose \( a, b, c, \in \mathbb{Z} \). If \( a^2 + b^2 = c^2 \), then \( a \) or \( b \) is even.
- Suppose \( a, b \in \mathbb{R} \). If \( a \) is rational and \( ab \) is irrational, then \( b \) is irrational.
- There exist no integers \( a \) and \( b \) for which \( 21a + 30b = 1 \).
- There exist no integers \( a \) and \( b \) for which \( 18a + 6b = 1 \).
- For every positive \( x \in \mathbb{Q} \), there is a positive \( y \in \mathbb{Q} \) for which \( y < x \).
- For every \( x \in [\pi/2, \pi] \), \( \sin x - \cos x \geq 1 \).
- If \( A \) and \( B \) are sets, then \( A \cap (B \setminus A) = \emptyset \).
- If \( b \in \mathbb{Z} \) and \( b \!\nmid\! k \) for every \( k \in \mathbb{N} \), then \( b=0 \).
- If \( a \) and \( b \) are positive real numbers, then \( a + b \geq 2 \sqrt{ab} \).
- For every \(n \in \mathbb{Z} \), \( 4 \!\nmid\! (n^2+2) \).
- Suppose \( a,b \in \mathbb{Z} \). If \( 4 \vert (a^2+b^2) \), then \( a \) and \( b \) are not both odd.

B. Prove the following statements using any method from Chapters 4, 5 or 6.

- The product of any five consecutive integers is divisible by 120. (For example, the product of 3, 4, 5, 6 and 7 is 2520, and \( 2520=120 \cdot 21\).)
- We say that a point \( P = (x,y) \in \mathbb{R}^2 \) is
**rational**if both \( x \) and \( y \) are rational. More precisely, \( P \) is rational if \( P \in \mathbb{Q}^2 \). An equation \( F(x,y)= 0 \) is said to have a**rational point**if there exists \( x_0, y_0 \in \mathbb{Q} \) such that \( F(x_0, y_0)=0 \). For ecample, the curve \( x^2+y^2-1=0 \) has rational point \( (x_0,y_0)=(1,0) \). Show that the curve \( x^2+y^2-3=0 \) has no rational points. - Exercise 20 (above) involved showing that there are no rational points on the curve \( x^2+y^2-3 \). Use this fact to show that \( \sqrt{3} \) is irrational.
- Use the above result to prove that \( \sqrt{3^k} \) is irrational for all odd, positive \( k \).
- The number \( \log_2 3 \) is irrational.