Question

Give the counterexample of the given conditional statement.

If $ab>0$then $a$and $b$are greater than 0.

Short answer

A counterexample to the given conditional statement is when $a=-3$and $b=-2$. The hypothesis is true; $\left(-3\right)\times \left(-2\right)=6>0$. However, the conclusion, $-3>0$and $-2>0$is false.

Step-by-step solution

Step 1. State the concept.

The product of two positive numbers and of two negative numbers are both positive.

Step 2. List the given data.

The given statement is “If $ab>0$, then, $a$and $b$are greater than 0.” Here, the hypothesis is “$ab>0$” and the conclusion is “$a$ and$b$ are greater than 0”.

Step 3. Simplify the given statement.

The given statement implies that if the product of two numbers is positive, then both these numbers are also positive. However, it is known that the product of two negative numbers is also positive.

Step 4. Give a counterexample.

Consider $a=-3$and $b=-2$.

Then,

$\begin{array}{l}ab=\left(-3\right)\left(-2\right)\\ =6\end{array}$

So, $ab>0$. Thus, the hypothesis is true.

However, $a=-3<0$and $b=-2<0$. This implies that the conclusion, “$a$ and $b$ are greater than 0” is false.

So, $a=-3$ and $b=-2$ is a counterexample of the given conditional statement.