Relation

In mathematics, a relation is any collection of ordered pairs. The fact that the pairs are ordered is important, and means that the ordered pair (a, b) is different from the ordered pair (b, a) unless a = b. For most useful relations, the elements of the ordered pairs are naturally associated or related in some way.

More formally, a relation is a subset (a partial collection) of the set of all possible ordered pairs (a, b) where the first element of each ordered pair is taken from one set (call it A), and the second element of each ordered pair is taken from a second set (call it B). A and B are often the same set; that is, A = B is common. The set of all such ordered pairs formed by taking the first element from the set A and the second element from the set B is called the Cartesian product of the sets A and B, and is written A × B. A relation between two sets then, is a specific subset of the Cartesian product of the two sets.

Since relations are sets at ordered pairs they can be graphed on the ordinary coordinate plane if they have ordered pairs of real numbers as their elements (real numbers are all of the terminating, repeating and nonrepeating decimals); for example, the relation that consists of ordered pairs (x, y) such that x = y is a subset of the plane, specifically, those points on the line x = y. Another example of a relation between real numbers is Figure 1. Illustration by Hans & Cassidy. Courtesy of Gale Group. the set of ordered pairs (x, y), such that x > y. This is also a subset of the coordinate plane, the half-plane below and to the right of the line x = y, not including the points on the line. Notice that because a relation is a subset of all possible ordered pairs (a, b), some members of the set A may not appear in any of the ordered pairs of a particular relation. Likewise, some members of the set B may not appear in any ordered pairs of the relation. The collection of all those members of the set A that appear in at least one ordered pair of a relation form a subset of A called the domain of the relation. The collection of members from the set B that appear in at least one ordered pair of the relation form a subset of B called the range of the relation. Elements in the range of a relation are called values of the relation. One special and useful type of relation, called a function, is very important. For every ordered pair (a, b) in a relation, if every a is associated with one and only one b, then the relation is a function. That is, a function is a relation for which no two of the ordered pairs have the same first element. Relations and functions of all sorts are important in every branch of science, because they are mathematical expressions of the physical relationships we observe in nature.