### Binary relations

The

*Cartesian product*AxB of two sets A and B is a set containing all of the ordered pairs of elements. For example, if A = {, } and B = {, }, then the Cartesian product is given by:

Another example consists of where the elements of AxB given that:

Notice that the number of elements in AxB is generally equal to the number of elements in A multiplied by the number of elements in B,

In the case where A and B are the same set, the Cartesian product AxA is the set containing all ordered pairs formed by the elements of A. For instance, if A = {1, 2, 3, 4} then:

AxA = {(1, 1), (1, 2), (1, 3), (1,4), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1), (3, 2), (3, 3), (3,4), (4, 1), (4, 2), (4, 3), (4, 4)}.

A *binary relation* on a set A is a subset of AxA. **The below video describes how relations and functions are applied to the Cartesian product**:

**Below is an activity detailing on the binary relations:**

**Applet courtesy of www.webspace.ship.edu**