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.”)


The expanses of the universe seem infinite to us, but in mathematics the concept of infinity reaches even beyond the edges of the universe toward numbers that are inconceivably large.


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