Answer to Question #198663 in Discrete Mathematics for Parth Godhani

Question #198663

What are the two types of indirect proofs? Explain through an example for each type.



1
Expert's answer
2021-06-08T04:33:43-0400

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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