1.1. Notation and Set Theory

Examples 1.1.5(b):

Prove that if the square of a number is an even integer, then the original number must also be an even integer. (Try a proof by contradiction).
To prove this we first need to know what exactly an even and odd integer is: Now that we have a precise definition, the actual proof is easy: Suppose x is a number such that x2 is even. To start a proof by contradiction we will assume the opposite of what we would like to prove: assume that x is odd (but x2 is still even). Then, because x is odd, we can write it as But then the square of x is with k = 2 n2 + 2n. Therefore x2 is odd. But that is contrary to our assumption that the square of x is even. Hence, if the square of x is supposed to be even, x itself must be 'not odd'. But 'not odd' means even. Therefore, the proof is finished.
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