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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.”)

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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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