5. Counter Examples

A counter example is an example that shows the hypothesis to be True but the conclusion to be false. As we have seen when you have a conditional statement the only time that the statement is false is when your hypothesis is True and your conclusion is False. So if you can show an example where the hypothesis is True and the conclusion False then the statement is not valid.

So let us look at an example of a counterexample question:

Statement:

All odd whole numbers between 1 and 11 are prime numbers

First off if you’ve forgotten what prime numbers are they are numbers that can only be divided by themselves and 1. So 3 is a prime number because 3x1 is the only way to get 3. 4 is not a prime number because you can get it by 2x2 and 4x1. Ok so let’s find a counterexample to this problem. First what are the hypothesis and the conclusion?

Hypothesis:

Odd whole numbers between 1 and 11

Conclusion

Are prime numbers

So now we have the hypothesis and conclusions first thing is to prove the hypothesis true so write out all the odd numbers between 1 and 11.

3,5,7,9

Now we look at the conclusion are all these numbers prime?

Well no if we take 9 we can get it by 3x3 and 9x1 so it is not a prime number. So since 9 is and odd number(hypothesis = True) and is not a prime number(Conclusion = False) this is a valid counterexample.

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