Recitation 4: Practice with Proofs

Part One

Your TA will guide you through the following proofs using different techniques.

Proof by Contrapositive

If x and y are integers and x - y is odd, then x is odd or y is odd.

Proof by Contradiction

Among any group of 25 people, there must be at least three who are all born in the same month.

Proof by Cases

If x is an integer, then x² + 5x - 1 is odd.

Part Two: Exercises

You will work individually through the following proofs.

1. There is no smallest integer.
2. For every real number x, if x is irrational, then -x is also irrational.
3. If x and y are real numbers, then max(x, y) + min(x, y) = x + y.
4. Let x and y be two integers. If xy is not an integer multiple of 5, then neither x nor y is an integer multiple of 5.
5. Every perfect square is either a multiple of 4 or a multiple of 4 plus 1.