__Domain, Co-domain and Range of a Relation__

A relation is a set of ordered pairs. Each ordered pair consists of two elements, called the "domain" and "range". The domain is the set of all first elements of the ordered pairs, while the range is the set of all second elements of the ordered pairs. The codomain is the set of all possible second elements of the ordered pairs.

- Domain of relation R (Dom(R) ) is the set of all those elements a ∈ A such that (a, b) ∈ R for some b ∈ B.
- If R be a relation from A to B, then B is the co-domain of R.
- Range of relation R is the set of all those elements b ∈ B such that (a, b) ∈ R for some a ∈ A.

In short: Domain = Dom(R) = {a : (a, b) ∈ R} and Range (R) = {b : (a, b) ∈ R}

**Note: **Range is always a subset of co-domain.

**Example** If f = {(5, 2), (6, 3)} and g = {(2, 5), (3, 6)}, write the range of f and g.

Solution The range of f = {2, 3} and the range of g = {5, 6}

# Composition of Relation

## Suppose that we have three sets A, B and C. A relation R defined from A to B, and a relation S defined from B to C. We can now define a new relation known as the composition of R and S, written as S o R. This new relation is defined as follows:

If a is an element in A and c is an element in C, then a(S o R)c, if and only if, there exists some element b in B, such that aRb and bSc. This means that, we have a relation S o R from a to c, if and only if, we can reach from a to c in two steps; i.e. from a to b related by R and from b to c related by S. In this manner, relation S o R can be interpreted as R followed by S, since this is the order in which the two relations need to be considered, first R then S.

**Calculating Composition of Relations:**

To compute S o R, we will first find the 2nd's of R (i.e., the b's) and let them be the 1st's (the a's) in S. In other words, R is computed 1st, then S i.e., it is right to left, not left to right.

**Properties of Composition of Relations:**

## 1. Let A, B, C and D be sets, R a relation from A to B, S a relation from B to C and T is a relation from C to D, then, T o (S o R) = (T o S) o R

## 2. Consider R be a relation from A to B and S is a relation from B to C. Then, if A is any subset of A, we have (S o R)(A) = S(R(A))

## 3. Let A, B and C are three sets, or a relation from A to B, and S a relation from B to C.

## Then, (S o R)^{-1} = R^{-1} o S^{-1}

## 4. A transitive closure of a relation R is the smallest transitive relation containing R, where a relation R is said to be transitive, if for every (a; b) belonging to R and (b; c) belonging to R, there is a (a; c) belonging to R.