What are the two types of indirect proofs? Explain through an example for each type.
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Proof of the Contrapositive:
The contrapositive of the statement P⇒Q is ¬Q⇒¬P.
Example: If ab is even then either a or b is even.
Assume both a and b are odd. Since the product of odd numbers is odd, ab is odd.
Proof by Contradiction:
To prove a sentence P by contradiction we assume ¬P and derive a statement that is known to be false.
Example: There are infinitely many primes.
Assume there are only finitely many primes . Let . Since , n is divisible by some prime, say . Then , so .
Since , there is a contradiction .