Relations

RELATIONS: Relationship between elements of sets is represented using a mathematical structure called relation. The most intuitive way to describe the relationship is to represent in the form of ordered pair. In this section, we study the basic terminology and diagrammatic representation of relation.
Definition : Let A and B be two sets. A binary relation from A to B is a subset of A X B.

Let A and B be two sets. Suppose R is a relation from A to B (i.e. R is a subset of A X B). Then, R is a set of ordered pairs where each first element comes from A and each second element from B. Thus, we denote

it with an ordered pair (a, b), where a ∈ A and b ∈ B. We also denote the relationship with a R b, which is read as a related to b. The domain of R is the set of all first elements in the ordered pair and the range of R is the set of all second elements in the ordered pair.

Example : Let A = { 1, 2, 3, 4 } and B = { x, y, z }. Let R = {(1, x), (2, x), (3, y), (3, z)}. Then R is a relation from A to B.

Example : Suppose we say that two countries are adjacent if they have some part of their boundaries common. Then, “is adjacent to”, is a relation R on the countries on the earth. Thus, we have, (India, Nepal) ∈ R, but (Japan, Sri Lanka) ∉ R.

Example : A familiar relation on the set Z of integers is “m divides n”. Thus, we have, (6, 30) ∈ R, but (5, 18) ∉ R.

Example : Let A be any set. Then A X A and Φ are subsets of A X A and hence they are relations from A to A. These are known as universal relation and empty relation, respectively.

Note : As relation is a set, it follows all the algebraic operations on relations that we have discussed earlier.

Definition : Let R be any relation from a set A to set B. The inverse of R, denoted by R^–1, is the relation from B to A which consists of those ordered pairs, when reversed, belong to R. That is:
R^–1 = {(b, a) : (a, b) ∈ R}

Example : Inverse relation of the relation in example 1.1 is, R^–1 = {(x, 1), (x, 2), (y, 3), (z, 3)}.