3.    Show that the following statement is true by the method of contrapositive.

                p: If x is an integer and x^2 is even, then x is also even.

Answers (1)
H Harsh Kankaria

Given, If x is an integer and x^2 is even, then x is also even. 

Let, p : x is an integer and x^2 is even

q: x is even

In order to prove the statement “if p then q” 

Contrapositive Method:  By assuming that q is false, prove that p must be false.

So,

q is false: x is not even \impies\implies x is odd \implies x = 2n+1 (n is a natural number)

\\ \therefore x^2 = (2n+1)^2 \\ \implies x^2 = 4n^2 + 4n + 1 \\ \implies x^2 = 2.2(n^2 + n) + 1 = 2m + 1

Hence x^2 is odd \impliesx^2 is not even 

Hence p is false.

Hence the given statement is true.

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