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p -> q
Proof by contraposition
~q -> ~ p (not q implies not p), means original is true
Proof by Contradiction
1) Assume opposite of hypotheses is true
2) Show that not p implies q is true
3) Demonstrates that q is false
4) Then not q is true.
5) Contraposition of 2. ~ q -> ~(~p)
6) ~ q -> ~ p
7) Therefore, p is true
Proof by Contradiction example
Suppose that we want to prove that root 2 is irrational.
NOTE: If it is a rational number, you can write it is a fraction.
In translation, it could be written as x = m / n (cannot write an irrational number in this form)
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