Recitation 3: Practice with Proofs
Part One
Your TA will guide you through the following proofs using different techniques.
Direct Proof
Show that the sum of two odd integers is even.
Proof by Contrapositive
If and are integers and is odd, then is odd or 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 is an integer, then is odd.
Part Two
Now work in small groups to prove the following propositions:
- Show that the sum of two even integers is even.
- If an integer is not divisible by 2, then it is not divisible by 4.
- The square of any integer is either a multiple of 4 or a multiple of 4 plus 1.
- If and are real numbers, then .
- There is no smallest integer.
- is irrational (not representable by a fraction).