1. Indirect Proof by Contradiction Direct Proof by Cases
Sections 2.4 and 2.5
Indirect Proof by Contradiction
Direct Proof by Cases

2. Review: Proof by contraposition
Express the statement in the form:
 x , if P(x) then Q(x)
Rewrite this statement in the contrapositive form:
 x , if not Q(x) then not P(x)
Prove the contrapositive in a direct proof

3. 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?

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

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

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

7. Section 2.5
Proof by cases

8. Proof by Cases
Sometimes a general proof over ALL cases is nearly impossible to do.
But, the proof can be clearly divided into several distinct cases which cover all possible situations.
The proof then consists of several smaller proofs where you cover each case individually.

9. Examples of this technique
The product of any two consecutive integers is even.

10. Examples of this technique
Prove that if a and b are real numbers, where neither is equal to zero, then  

11. Examples of this technique
For all real numbers x,

12. Your Turn Contradiction Cases If 3n+2 is odd, then n is odd
[Hint, be careful writing the negation of this statement].
Cases
Let n be an integer. If 3 does not divide n, then 3 divides n2−1.

13. If Time:Proof by Contrapositive
If x and y are two integers for which x+y is even, then x and y have the same parity.
DEFINITION - Two integers are said to have the same parity if they are both odd or both even.