Arelation from a set x to a set y is called a function if each element of x is related to exactly one element in y. that is, given an element x in x, there is only one element in y that x is related to. for example, consider the following sets x and y.

An ordered pair is a set of inputs and outputs and represents a relationship between the two values. A relation is a set of inputs and outputs, and a function is a relation with one output for each input.

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here is a exstended explantion:

A relation from a set X to a set Y is called a function if each element of X is related to exactly one element in Y. That is, given an element x in X, there is only one element in Y that x is related to.

For example, consider the following sets X and Y. I'll give you a relation between them that is not a function, and one that is.

   X = { 1, 2, 3 }

   Y = { a , b , c, d }

   Relation from X to Y (i.e., in XxY ) : { (1,a) , (2, b) , (2, c) , (3, d) }

This relation is not a function from X to Y because the element 2 in X is related to two different elements, b and c. (Note, if you transpose the ordered pairs you <i>would</i> have a function from Y to X - can you see WHY?)

   Relation from X to Y that is a function: { (1,d) , (2,d) , (3, a) }

This is a function since each element from X is related to only one element in Y. Note that it is okay for two different elements in X to be related to the same element in Y. It's still a function, it's just not a one-to-one function.