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Question

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.

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Solution

The given statement is: p: “If x is an integer and x 2 is even, then x is also even.” To show that p is true by the method of contrapositive, first break the given statement into parts as follows:

q: x is an integer and x 2 is even.

r: x is even.

Consider that r is false then show that q must be false. Since the statement r is false, then it should be the negation of the statement r which will be denoted as r .

r : x is not even.

x is not even implies that its square is also not even. Therefore the statement q is not true.

Thus, the given statement p is true.


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