Question

Decide whether the statement is true or false . If it is false, give a counterexample. The product \((-a) \cdot(-1)\) is always positive.

Short answer

The statement is false. The counterexample is when a = 0. The product in this case is \( 0 \cdot -1 = 0 \), which is not positive.

Step-by-step solution

Understand the Statement

First, it's crucial to understand what the statement means. The statement is saying \( -a \cdot -1 \) is always positive. Let's check if it's true or false.

Test the Statement using an Example

Let's take \( a = 2 \). Now, \( -a = -2 \). Thus, substituting the value of \( -a \) in \( -a \cdot -1 \), we get \( -2 \cdot -1 = 2 \) which is indeed positive.

Generalize the Result

By introducing any number in place of \( a \), one can see that \( -a \cdot -1 = a \), which is the absolute value of \( a \) and is always positive when \( a \) is not equal to zero. However, if \( a = 0 \), then \( -a \cdot -1 = 0 \cdot -1 = 0 \), which is neither positive nor negative. Hence the statement is false.

Provide Counterexample

A counterexample is a case where the statement doesn't hold true. An example of this is when \( a = 0 \). In this case \( -a \cdot -1 = 0 \), which is not positive, thus making the statement false.