Relations

Let there be two sets A and B. Then, any subset of a Cartesian product A×B of two sets A and B is called a relation.

A relation from a set A to a set B is denoted by xℜy, if x∈A and y∈B, or simply by ℜ if (x, y)∈ℜ. A relation from a set A to itself is called a relation on A.

Let A={1, 2, 3} and B={1, 2, 3, 4}, then,

A×B={(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1), (3, 2), (3, 3), (3, 4)}

Then, a relation ℜ from A to B be described by ℜ={(x, y): x=y} is ℜ={(1, 1), (2, 2), (3, 3)} ⊂ A×B.

Another example, ℜ1={(x, y):x<y} is ℜ1={(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)} ⊂ A×B.

In tabular form, the above relations can be represented as;

Tabular form representing relations

In graph, the above relations can be represented as;

Graph representing relations

Domain and Range

The domain of a relation ℜ is the set of all first members of the pairs (x, y) of ℜ. It is denoted by Dom(ℜ).

Dom(ℜ)={x:(x, y)∈ℜ for some y∈B}

The range of a relation ℜ is the set of all second members of the pairs (x, y) of ℜ. It is denoted by Ran(ℜ).

Ran(ℜ)={y:(x, y)∈ℜ for some x∈A}

Example 1. ℜ={(1, 1), (2, 2), (3, 3)}

Dom(ℜ)={1, 2, 3} and Ran(ℜ)={1, 2, 3}

Example 2. ℜ1={(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)}

Dom(ℜ1)={1, 2, 3} and Ran(ℜ1)={2, 3, 4}

Arrow diagrams representing relations

Inverse Relation

Since a relation is a subset of a Cartesian product, a set can be formed by interchanging the first and second members of a relation. Thus, every relation from a set A to a set B has a relation from B to A. Suppose a relation,

ℜ={(x, y):x∈A, y∈B}⊂A×B

then, a relation of the form,

{(y, x):y∈B, x∈A}⊂B×A is called inverse relation from B to A of ℜ. This relation is denoted by ℜ-1.

Let A={1, 2, 3} and B={a, b} be two sets. Then,

A×B={(1, a), (1, b), (2, a), (2, b), (3, a), (3, b)}

and, B×A={(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)}

If ℜ={(1, a), (2, b)}, then ℜ-1={(a, 1), (b, 2)}

Also, Dom(ℜ)={1, 2} and Ran(ℜ)={a, b}

and, Dom(ℜ-1)={a, b} and Ran(ℜ-1)={1, 2}

Again, let’s consider relation on A,

A×A={(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}. Here, the elements (1, 1), (2, 2) and (3, 3) are called the diagonal elements of A×A.

If ℜ={(1, 1), (2, 2), (3, 3)} then ℜ-1={(1, 1), (2, 2), (3, 3)}

Dom(ℜ)={1, 2, 3}=Dom(ℜ-1)

Ran(ℜ)={1, 2, 3}=Ran(ℜ-1)

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