1.30 Relation and Function Definitions

Relation and Function Definitions establishes the vocabulary for describing correspondences between two sets of values, distinguishing the general case in which values may correspond to more than one partner from the more restricted case in which each value corresponds to exactly one, and identifying the specific roles that the two members of each corresponding pair play.

Relation

A relation is any set of ordered pairs connecting values from one set to values from another, without restriction on how many times a given first value may appear paired with different second values. The set of ordered pairs (1, 2), (1, 5), (3, 4) is a relation, even though the value 1 appears paired with two different second values, since a relation places no requirement on uniqueness of correspondence.

Function

A function is a relation in which every first value corresponds to exactly one second value, with no first value ever paired with more than one distinct second value. The set of ordered pairs (1, 2), (3, 4), (5, 4) is a function, since each first value—1, 3, and 5—appears exactly once, even though the second value 4 is repeated; a repeated second value does not violate the requirement, only a repeated first value paired with different second values would.

Input

The input of a relation or function is the first value of a given ordered pair, representing the value supplied to the relationship before its corresponding partner value is determined. In the ordered pair (3, 4), the input is 3, and in a symbolic function written as f(x), the input is the value substituted for x.

Output

The output of a relation or function is the second value of a given ordered pair, representing the value that results from, or corresponds to, a specific input. In the ordered pair (3, 4), the output is 4, and in a symbolic function written as f(3) = 4, the output is the resulting value 4 produced by supplying the input 3.

Together, these definitions establish the function as a specific, more restricted type of relation—one guaranteeing a single, unambiguous output for every input—providing the foundational vocabulary for describing correspondences between values and for later distinguishing which everyday and algebraic relationships qualify as functions and which do not.