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# Relation, and its types - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

## Quick Facts

• Reflexive, Symmetric and Transitive relation, Equivalence relation are considered the most difficult concepts.

• 55 Questions around this concept.

## Solve by difficulty

Let $\dpi{100} R$ be the real line. Consider the following subsets of the plane $\dpi{100} R \times R$ :

$\dpi{100} S=\left \{ (x,y):y=x+1\; and\; 0< x< 2 \right \}$

$\dpi{100} T=\left \{ (x,y):x-y\; is\;an\; integer \right \}$

Which one of the following is true?

Let P be the relation defined on the set of all real numbers such that

$\dpi{100} P=\left \{ \left ( a,b \right ) :sec^{2}a-\tan ^{2}b=1\right \}$. Then P is

Let $S=\{1,2,3, \ldots, 10\}$. Suppose $M$ is the set of all the subsets of $S$, then the relation $\mathrm{R}=\{(\mathrm{A}, \mathrm{B}): \mathrm{A} \cap \mathrm{B} \neq \phi ; \mathrm{A}, \mathrm{B} \in \mathrm{M}\}$ is :

Let $R$ be a relation on $Z \times Z$ defined by $(a, b) R(c, d)$ if and only if $a d-b c$ is divisible by 5 .The R is

If $\mathrm{R}$ is the smallest equivalence relation on the set $\{1,2,3,4\}$ such that $\{(1,2),(1,3)\} \subset \mathrm{R}$, then the number of elements in $\mathrm{R}$ is________.

## Concepts Covered - 3

Universal relation, Empty Relation, Identity relation

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 IA

R = {(a, b) : a ∈ A, b ∈ A and a = b}

It can also be written as $\\\mathrm{I_A=\left \{ (a,a)\;:\;a\in A \right \}}$

For example A = {2 ,4, 6}

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

Reflexive, Symmetric and Transitive relation

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}

• R= {(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, Ris 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
• R= {(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)}
• R= {(2,3),(3,1)}
• R= {(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 R is a transitive relation.

Equivalence relation

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

## Study it with Videos

Universal relation, Empty Relation, Identity relation
Reflexive, Symmetric and Transitive relation
Equivalence relation

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