## Foundations of Mathematics## Set Theory |

## How do functions pertain to sets? |

A function in sets pertains to a correspondence between two sets, called the *domain* and *range;* each member of the domain has exactly one member of the range. It is often called a many-to-one (or sometimes one-to-one) relation. For example, f = {(1,2), (3,6), (4, -2), (8,0), (9,6)} is a function, with each set of numbers being ordered pairs. This is because it assigns each member of the set {1, 3, 4, 8, 9} exactly one value in the set {2, 6, -2, 0, 6}. It never has two ordered pairs with the same *x* and different *y* values. In this case, the domain is {1, 3, 4, 8, 9} and the range is {2, 6, -2, 0, 6}.

To show an example that is *not* a function, f = {(1,8), (4,2), (3,5), (1,3), (6,11)} is *not* a function because it does not assign each member of the set exactly one value: It assigns *x* = 1 to both *y* = 8 and *y* = 3, or it has two ordered pairs that have the same *x* values to two different *y* values, (1, 8) and (1, 3). (For more information about functions, see “Algebra.”)