# 3.4: Functions

Both the real world and the world of mathematics are full of what are called, in math- ematics, ‘functional relationships’. A functional relationship is a relationship between two sets, which associates exactly one element from the second set to each element of the first set.

For example, each item for sale in a store has a price. The first set in this relationship is the set of items in the store. For each item in the store, there is an associated price, so the second set in the relationship is the set of possible prices. The relationship is a functional relationship because each item has a price. That is, the question “What is the price of this item?” has a single, definite answer for each item in the store.

Similarly, the question “Who is the (biological) mother of this person?” has a single, definite answer for each person. So, the relationship ‘mother of’ defines a functional relationship. In this case, the two sets in the relationship are the same set, namely the set of people. On the other hand, the relationship ‘child of’ is not a functional relationship. The question “Who is the child of this person?” does not have a single, definite answer for each person. A given person might not have any child at all. And a given person might have more than one child. Either of these cases—a person with no child or a person with more than one child—is enough to show that the relationship ‘child of’ is not a functional relationship.

Or consider an ordinary map, such as a map of Zuid-Holland or a street map of Rome. The whole point of the map, if it is accurate, is that there is a functional relationship between points on the map and points on the surface of the Earth. Perhaps because of this example, a functional relationship is sometimes called a mapping.

There are also many natural examples of functional relationships in mathematics. For example, every rectangle has an associated area. This fact expresses a functional relationship between the set of rectangles and the set of numbers. Every natural numbern has a square, n2. The relationship ‘square of’ is a functional relationship from the set of natural numbers to itself.