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3.5 Counterexamples

If we want to establish that a statement is true, it is not sufficient simply to check particular cases: we may choose luckily and miss the cases where the statement fails.

Example 3.5.1

Consider the statement ‘‘all numbers are rational’’. If you did not know Example 3.4.2 and just chose to check the statement against the first few numbers which came to mind, you might end up thinking that it is true: $1$ is rational, $2$ is rational, $1/2$ is rational, $-7/22$ is rational … However, the statement is false as Example 3.4.2 shows: $\sqrt{2}$ is an example of a number which is not rational.

In general, to show that a given statement is false, it suffices to find a single example where it fails. We refer to such an example as a counterexample to the statement. Thus $\sqrt{2}$ is a counterexample to the statement considered in Example 3.5.1.

Example 3.5.2

For integers $a$ , $b$ and $c$ , consider the statement ‘‘if $b>c$ , then $ab>ac$ ’’. Although this is true for some choices of $a$ , $b$ and $c$ , it is not true for all choices. For example, if $a=-1$ , $b=2$ and $c=1$ , then $b=2>1=c$ , but $ab=-2<-1=ac$ , and therefore the statement is false in general.