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.