Relations — Domain and Range

A relation $R$ from a set $A$ to a set $B$ is simply any subset of the Cartesian product $A \times B$, i.e. $R \subseteq A \times B$. In plain terms, a relation is a chosen collection of ordered pairs that connects some elements of $A$ to some elements of $B$ according to some rule or description — 'is the price of', 'is the manager of', 'is greater than', and so on.

Two sets are naturally associated with a relation:

• The domain of $R$ is the set of all first elements of the ordered pairs in $R$ — i.e. the elements of $A$ that actually appear paired with something.
• The range of $R$ is the set of all second elements of the ordered pairs in $R$ — i.e. the elements of $B$ that actually appear as an image.

Note carefully that the domain need not be the whole of $A$, and the range need not be the whole of $B$ — a relation may simply leave some elements of $A$ unpaired. This is the crucial point that separates a general relation from a function, taken up in the next section. …