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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 p_1,...,p_k . Let n=p_1...p_k+1 . Since n\ge2 , n is divisible by some prime, say p_i . Then p_i|(p_1...p_i...p_k) , so p_i|(n-p_i...p_k) .

Since n-p_i...p_k , there is a contradiction p_i|1 .


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