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A relation between two sets is a collection of ordered pairs containing one object from each set. If the object $x$ is from the first set and the object $y$ is from the second set, then the objects are said to be related if the ordered pair $(x,y)$ is in the relation.

A function is a type of relation. But, a relation is allowed to have the object $x$ in the first set to be related to more than one object in the second set. So a relation may not be represented by a function machine, because, given the object $x$ to the input of the machine, the machine couldn't spit out a unique output object that is paired to $x$.