A subset of the Cartesian product of two sets is called a relation. If the relation between sets A and B is denoted by R, then R ⊆ A × B. The subset R is defined according to the relation between the first and second elements of A × B. The second element of each ordered pair is called the image of the first element in the relation R.

From one single A × B, many different relations can be formed, depending on the rule used to pick pairs — "is less than", "is equal to", "is greater than", and so on. Because relations can behave very differently, they are grouped into types of relation — reflexive, symmetric, transitive and equivalence — based on the properties they satisfy. See the example below.