Topic 1.6 Logic and Proof II - Contrapositive and Contradiction

Question 1

Prove the following statement with a proof by contrapositive. Suppose x ϵ Z. If 7x + 9 is even, then x is odd

Answer 1

To prove this, suppose x is even. Then, 7x is even and so 7x+9 is odd

Question 2

What does it mean to reach a contradiction?

Answer 2

It means to reach a conclusion that is both true and false
If you suppose p and then you reach something impossible, you get to rewind to your hypothesis and deduce that you proved the opposite. p can’t be possible because it would lead to a contradiction, so it must be that not p is true