Relation, and its types - Practice Questions & MCQ

Relation on a set

If a relation is from A to A itself, then this relation is called relation on set A.

Empty Relation

A relation R on a set A is called empty relation, if no element of A is related to any element of A, i.e., R = φ

For example,  Let A = {2, 4, 6} and R = {(a, b) : a, b ∈ A and a + b is odd} 

Here, R contains no element, therefore R is an empty relation on A 

Universal Relation

A relation R on a set A is called universal relation, if each element of A is related to every element of A, i.e., R has all the ordered pair contained in A x A

So, R = A × A.

For example,

1. Let A = {2 ,4} and R = {(2,2), (2,4), (4,2), (4,4)}

  Here, R = A x A. Hence, R is Universal relation

2. Let A = {1,2,3}, and R = {(a, b) : |a - b| > -2, a , b ∈ A}

Clearly mod value of difference of any pair (a,b) will be greater than -2

So, each possible ordered pair in A x A will lie in R, therefore R is a universal relation

Identity relation

If every element of A is related to itself only, then it is known as an identity relation on A . It is denoted by I_A

R = {(a, b) : a ∈ A, b ∈ A and a = b} 

It can also be written as 

For example A = {2 ,4, 6}

Then, I_A = {(2,2), (4,4), (6,6)}

Reflexive Relation

A relation R on a set A is called Reflexive, if (a, a) ∈ R, for every a ∈ A,

For example: let A={1,2,3}

• R₁ = {(1,1), (2,2), (3,3)}
• R₂ = {(1,1),(2,2),(3,3), (1,2), (2,1),(1,3)}
• R₃ = {(1,1),(2,2), (2,3),(3,2)}

Here R₁ and R₂ are reflexive relations on A, R₃ is not a reflexive relation on A as (3,3) is not present in R₃.

Note: In identity relation, all elements of type (a.a) should be there and there should not be any other element. But in reflexive relation, all elements of type (a,a) should be there, and apart from these, other elements can also be there

So, R₁ is identity relation and is also reflexive, but  R₂ is only reflxive and not an identity relation

Symmetric Relation

A relation R on a set A is said to be symmetric relation, if a R b ⇒ b R a,∀ a,b ∈ A

For example, A={1,2,3}

• R₁ = {(1,2), (2,1)}
• R₂ = {(1,2), (2,1), (1,3),(3,1)} and
• R₃ = {(2,3),(1,3),(3,1)}

Here R₁ and R₂ are symmetric relations on A but R₃ is not a symmetric relation on A because (2,3) is in R₃ and (3,2) is not in R₃.

Transitive Relation

A relation R on a set A is said to be a transitive relation, if a R b and b R c ⇒ a R c, ∀ a,b,c ∈A

For example, Let  A={1,2,3}

• R₁ = {(1,2), (2,3),(1,3),(3,2)}
• R₂ = {(2,3),(3,1)}
• R₃ = {(1,3),(3,2),(1,2)}

Here R₁ is not a transitive relation on A because (2,3) is in R₁ and (3,2) is in R₁ but (2,2) is not in R₁ . Also, (3,2) in R₁ and (2,3) is in R₁, but (3,3) is not in R₁

Again R₂ is not transitive relation on A because (2,3) is in R₂ and (3,1) is in R₂ but (2,1) is not in R₂.

Finally R₃  is a transitive relation.

 A relation R on a set A is said to be an equivalence relation if R is reflexive, symmetric and transitive.