1. Direct Proof by Contraposition Direct Proof by Contradiction
Section 2.3
Direct Proof by Contraposition
Direct Proof by Contradiction

2. How would you prove
Prove: For all integers n, if n2 is even then n is even.

3. Remember the Definition of Contrapositive
Original Statement
p  q
Contrapositive
¬ q  ¬ p
If the original statement is true, so is the contrapositive.

4. Directions for Writing Proofs by Contraposition
Express the original statement to be proved in the form:
For all x, if A(x) is true then B(x) is true.
Rewrite this statement in the contrapositive.
For all x, if B(x) is false then A(x) is false.
Prove the contrapositive by direct proof.

5. Group Example
Prove: For all integers n if n2 is even then n is even

6. Group Examples
Prove: If a product of two positive real numbers is greater than 100, then at least one of the numbers is greater than 10.

7. Your turn
Prove: For all integers n if n2 is odd then n is odd. Prove: If the sum of two real numbers is less than 50, then at least one of the numbers is less than 25.

8. Proof by Contradiction
Another tool for “direct proof” is Proof by Contradiction.
Introduction:
If A is true, what do you know about ¬ A ?
If A is false, what do you know about ¬ A ?
If ¬ A is false, what do you know about A?

9. Proof by Contradiction
Suppose that the statement is false. That is, suppose the negation of the statement is true
Show that this new supposition leads to a contradiction.
Conclude that the original statement must be true.

10. Examples of this technique
There is no greatest integer

11. Examples of this technique
If a is a rational number and b is an irrational number, then a+b is an irrational number.

12. Examples of this technique
There is no integer that is both even and odd

13. Examples of this technique
If 3n+2 is odd, then n is odd
[Hint, be careful writing the negation of this statement].