Example 0.4.2.

Just because you can describe a rule in the same way you would write a function, does not mean that the rule is a function. The following are NOT functions.

1. \(f:\N \to \N\) defined by \(f(n) = \frac{n}{2}\text{.}\) The reason this is not a function is because not every input has an output. Where does \(f\) send 3? The rule says that \(f(3) = \frac{3}{2}\text{,}\) but \(\frac{3}{2}\) is not an element of the codomain.

2. Consider the rule that matches each person to their phone number. If you think of the set of people as the domain and the set of phone numbers as the codomain, then this is not a function, since some people have two phone numbers. Switching the domain and codomain sets doesn't help either, since some phone numbers belong to multiple people (assuming some households still have landlines when you are reading this).