In each of these problems, give a proof using the Method of Contradiction. Also give a direct proof if you can.

1. Problem 7a in Section 1.4 of the text.

2. Prove that  is irrational.

3. Prove that if m is an integer and m² is even, then m is even.

4. Prove there do not exist integers m and n such that 3m + 9n = 80.

5. Prove that if 100 letters are placed into 9 mailboxes, then some box contains 12 or

more letters.

6. Suppose x and y are real numbers. Prove that if x + y ? 2, then either x ? 1 or

y ? 1.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

PROOFS USING CONTRADICTION
 
 

In each of these problems, give a proof using the Method of Contradiction. Also give a direct proof if you can.

1. Problem 7a in Section 1.4 of the text.

2. Prove that  is irrational.

3. Prove that if m is an integer and m² is even, then m is even.

4. Prove there do not exist integers m and n such that 3m + 9n = 80.

5. Prove that if 100 letters are placed into 9 mailboxes, then some box contains 12 or

more letters.

6. Suppose x and y are real numbers. Prove that if x + y ? 2, then either x ? 1 or

y ? 1.