Question

Use the method of direct proof to prove the following statements. Suppose \(x, y \in \mathbb{Z} .\) If \(x\) is even, then \(x y\) is even.

Short answer

As \( x \) is even, it can be written as \(2n\). Thus, when \( x \) is multiplied with any integer \( y \), the result is \(2(ny)\), which is also an even number. Therefore, the statement is proven true.

Step-by-step solution

Understanding the Problem

The given condition of the problem is that \( x \) is an even integer. This means that \( x \) can be expressed as \( 2n \), where \( n \) is also an integer. The target is to prove that the product of \( x \) and any other integer is even. A number is defined as even if it can be expressed as the product of 2 and another integer. The objective is to demonstrate that \( x \times y = 2n \times y \) is also in this format.

Applying the Definition of an Even Number

Starting with the assumption that \( x \) is an even number, \( x \) can be expressed as \( 2n \), where \( n \) is an integer. This is the definition of an even number.

Proving the Result is Even

Now, let's multiply \( x \) with some arbitrary integer \( y \), obtaining \( x \times y = 2n \times y \). Rearranging terms, this becomes \( 2(n \times y) \). This is again a product of 2 and another integer, thus fulfilling the definition of an even number. This shows that the product of \( x \) and any integer \( y \) is indeed even.