What is Disproof by Counterexample? – Examples & Videos

Note that ${\mathbb Z}$ is the set of all positive or negative integers. Finding an $a$ and $b$ such that $a\ne b$ but $a^2=b^2$, then the statement is disproved. Choosing any integer for $a$ and then choosing $b=-a$ will accomplish this. For example, let $a=4$ and $b=-4$. In this case $a^2=16$ and $b^2=16$ and so we have found an example where $a^2=b^2$ but $a\ne b$ and thus disproving the statement.

The tendency in this situation is to try various positive values of $p$ and $q$ until the statement isn’t true. However, the statement IS true for all positive values and so negative values of $p$ and $q$ must be tried. Bear in mind that you can only square root a positive number (without complex numbers) and so both $p$ and $q$ must both be negative. Try, for example, $p=-2$ and $q=-2$. The left hand side is the square root of $pq=4$ which is $\pm 2$ whereas the right hand side is $-2$. Since 2 is not less than or equal to $-2$, we have found a counterexample.