First, let’s refresh the concept of direct proof.

“A direct proof of a conditional statement p → q is constructed when the first step is the assumption that p is true; subsequent steps are constructed using rules of inference, with the final step showing that q must also be true. A direct proof shows that a conditional statement p → q is true by showing that if p is true, then q must also be true, so that the combination p true and q false never occurs.” Discrete Mathematics and its Applications by Rosen.

Let’s rewrite this exercise in the same way as the definition of direct proof.

Let’s p=”n is an even number” and q=”the square of n is an even number”.

We must proof p->q.

We should assume that p is true and take subsequent steps, using rules of inference, to show at the end that q is also true.

If p is true, meaning n is an even number, then n=2k for some integer k (definition of an even number).

n^{2}=(2k)^{2}=4k^{2}=2(2k^{2})

By the definition of even number, the square of an even number is also even.

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**Related exercises:**

- 4. Show that the additive inverse, or negative, of an even number is an even number using a direct proof
- 5. Prove that if m + n and n + p are even integers, where m, n, and p are integers, then m + p is even. What kind of proof did you use?
- 6- Use a direct proof to show that the product of two odd numbers is odd
- 7. Use a direct proof to show that every odd integer is the difference of two squares.