# Types of Relations

## Types of Relations

There are various types of relations between two sets. Here are some of the most common types of relation

**Empty Relation**

An empty relation (or void relation) is one in which there is no relation between any elements of a set. **For example,** if set A = {1, 2, 3} then, one of the void relations can be R = {x, y} where, |x – y| = 8. For empty relation,

R = φ ⊂ A × A

**Universal Relation**

A universal (or full relation) is a type of relation in which every element of a set is related to each other.

Consider set A = {a, b, c}. Now one of the universal relations will be R = {x, y} where, |x – y| ≥ 0. For universal relation,

R = A × A

**Identity Relation**

In an identity relation, every element of a set is related to itself only.

**For example**, in a set A = {a, b, c}, the identity relation will be I = {a, a}, {b, b}, {c, c}. For identity relation,

I = {(a, a), a ∈ A}

**Inverse Relation**

Inverse relation is seen when a set has elements which are inverse pairs of another set.

**For example** if set A = {(a, b), (c, d)}, then inverse relation will be R^{-1} = {(b, a), (d, c)}. So, for an inverse relation,

R^{-1} = {(b, a): (a, b) ∈ R}

**Reflexive Relation**

In a reflexive relation, every element maps to itself. For example, consider a set A = {1, 2,}. Now an example of reflexive relation will be R = {(1, 1), (2, 2), (1, 2), (2, 1)}. The reflexive relation is given by-

(a, a) ∈ R

**Symmetric Relation**

In a symmetric relation, if a=b is true then b=a is also true. In other words, a relation R is symmetric only if (b, a) ∈ R is true when (a,b) ∈ R.

An example of symmetric relation will be R = {(1, 2), (2, 1)} for a set A = {1, 2}. So, for a symmetric relation,

aRb ⇒ bRa, ∀ a, b ∈ A

**Transitive Relation**

For transitive relation, if (x, y) ∈ R, (y, z) ∈ R, then (x, z) ∈ R. For a transitive relation,

aRb and bRc ⇒ aRc ∀ a, b, c ∈ A

**Equivalence Relation**

If a relation is reflexive, symmetric and transitive at the same time it is known as an equivalence relation.

** Note:** An important property of an equivalence relation is that it divides the set

into pairwise disjoint subsets called equivalent classes whose collection is called

a partition of the set. Note that the union of all equivalence classes gives

the whole set

** Example:- ** Let us assume that F is a relation on the set **R** real numbers defined by xFy if and only if x-y is an integer. Prove that F is an equivalence relation on **R.**

**Solution:**

**Reflexive**: Consider x belongs to **R**,then x – x = 0 which is an integer. Therefore xFx.

**Symmetric**: Consider x and y belongs to **R** and xFy. Then x – y is an integer. Thus, y – x = – ( x – y), y – x is also an integer. Therefore yFx.

**Transitive**: Consider x and y belongs to **R**, xFy and yFz. Therefore x-y and y-z are integers. According to the transitive property, ( x – y ) + ( y – z ) = x – z is also an integer. So that xFz.

Thus, R is an equivalence relation on **R**.