Types of Relations

Many mathematical models are relations, such as the relation of equality, the relation of equivalence, the relation of order, etc. Relations in mathematics describe a way in which two mathematical expressions are related to each other. The four basic types of relations in math are equality, less than, greater than and not equal to. Relations can describe different functional relationships.

Relations are important to set theory because they relate sets and functions. Relations, functions and sets are all related concepts of each other.

Let us first start with the definition of set and the relation,  If S be  a set or a collection of objects and x a member of S  that is if x is an object which belongs to S, we shall write x ∈ S and say that” x is a n element of S “or  that “x is a member of S “

Here, we will learn about what relations are and some types of relations. Relations connect two or more sets. There are many types of relations which are listed below.

• Empty Relation

• Universal Relations

• Identity Relations

• Inverse Relations

• Reflexive Relations

• Symmetric Relations

• Transitive Relations

• Equivalence Relations

Let’s discuss each of these relation types

Empty Relations

 When there is no relation between any elements of a set then it is referred to as Empty relation or Void relation

If set A = {1, 2, 3} then, one of the empty relations could be R = {x, y} where, |x – y| = 8. 

For empty relation,

R = φ ⊂ A × A

Universal Relations

The relation where all the elements of a given set are related to each other.

A = {a, b, c}. Now let’s take an example of  one of the universal relations and that will be R = {x, y} where, |x – y| ≥ 0. 

Hence, R = A × A

Identity Relations

when every element of a set is related to itself only then it is known as identity relations. 

 Let us take a relation, A = {a, b, c}, the identity relation can  be= {a, a}, {b, b}, {c, c}.

Thus, I = {(a, a), a ∈ A}

Inverse Relation

Inverse relation occurs when a set have elements better inverse in pairs

A = {(a, b), (c, d)}, then the inverse relation can be R-1 = {(b, a), (d, c)}

Here, R-1 which is equal to {(b, a): (a, b) ∈ R}

Reflexive Relation

In a binary relation on   a set X, when every element of X relates to itself then it is known as reflexive relation. 

R = { (8, 8), (7, 7), (9, 9)}

(a, a) ∈ R

Symmetric Relation

We know that, if a =b then it is also true that b=a. That means the given relation R is symmetric only if (b, a) ∈ R is true when (a,b) ∈ R. 

So we can write, R = {(1, 2), (2, 1)} for a set A = {1, 2}.

aRb is equal to  bRa, ∀ a, b ∈ A

Transitive Relation

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

xRy and yRz ⇒ xRz ∀ z, y, z ∈ A

Equivalence Relation

If the relation is reflexive, symmetric and transitive at the same time then, it is called an equivalence relation.