
Up to this point we have used direct proof or counterexamples. In a direct proof we start with the hypotheses and make deductions until we reach a conclusion.
In an indirect proof we prove some statement other than the statement of the theorem. Then we use logic to show the theorem follows from the statement actually proved. The two methods of proof that we will look at are:
1. Proof by contradiction
2. Proof by contraposition
Contradiction
1. Assume that the statement is false.
2. Show that this leads logically to a contradiction
3. Conclude the statement to be proved is true. ex. Prove:
If x

R

, x x

1
 x x

1

2 .

ex. Prove: If m is an even integer, then m + 7 is odd.
Prove: The negative of an irrational number is irrational.
Do: Prove: For all integers n, if n 2 is odd, then n is odd.



Contraposition
1. The statement is in the form
 x

D , if P ( x ), then
2. Rewrite the statement as a contrapositive
Q ( x ).
 x

D , if ~ Q ( x ), then ~ P ( x ).
3. Prove the contrapositive by direct proof. ex. If m is an even integer then m + 7 is odd. ex. If x ≠ y, then e x ≠ e y .


Sometimes it’s easier to prove the contrapositive. For instance we can make the statement use rationals instead of irrationals or we can make the statement less complicated. ex.
 m , n

Z

and m
 n , if m
2  n
2
, then m
 n .
ex. The negative of an irrational number is irrational.

If x is an irrational number, then –x is irrational.
Do: Prove: For all integers n, if n 2 is odd then n is odd.