Let R be the real line. Consider the following subsets of the plane R × R S = { ( x , y ) ∣ y = x + 1 , 0 < x < 2 } T = { ( x , y ) ∣ x - y is an integer } . Which one of the following is true?

T is an equivalence relation on R but S is not

Neither S nor T is an equivalence relation on R

Both S and T are equivalence relation on R

S is an equivalence relation on R and T is not

Let the relation $\rho$ be defined on R as a ρ b if 1 + a b > 0 . Then

ρ is reflexive and symmetric but not transitive.

ρ is reflexive and transitive but not symmetric

Let Z denote the set of all integers. If a relation R is defined on Z as follows: ( x, y) ∈ R if and only if x is multiple of y, then R is

reflexive, transitive but not symmetric

reflexive, symmetric but not transitive

symmetric, transitive but not reflexive

neither reflexive nor transitive but symmetric

Let R 1 and R 2 be two relations defined as follows : R 1 = a , b ∈ R 2 : a 2 + b 2 ∈ Q and R 2 = a , b ∈ R 2 : a 2 + b 2 ∉ Q , where Q is the set of all rational numbers, then

Neither R 1 nor R 2 is transitive.

R 1 is transitive but R 2 is not transitive.

R 2 is transitive but R 1 is not transitive.

R 1 and R 2 are both transitive.

Let R be the real line. Consider the following subsets of the plane R × R S = { x , y : y = x + 1 and 0 < x < 2 } T = { x , y : x - y is an integer} Which of the following is true?

T is an equivalence relation on R but S is not

Neither S nor T is an equivalence relation on R

Both S and T are equivalence relations on R

S is an equivalence relations on R and T is not

Show that the relation R on R is defined as R = { ( a , b ) : a ≤ b } , is reflexive and transitive but not symmetric.

Given, R = a , b : a ≤ b , a , b ∈ R For the relation R to be reflexive: If the relation R is reflexive, then a , a ∈ R i.e. a ≤ a , which is always true for all the values of a , a ∈ R So, the given relation R is reflexive. For the relation R to be symmetric: If a , b ∈ R then b , a ∈ R i.e. if a ≤ b then b ≤ a , which is not possible for all the values of a , b ∈ R . For example: if a = 2 and b = 5 Here, a < b but b > a So, the given relation R is not symmetric. For the relation R to be transitive: If a , b ∈ R and b , c ∈ R then a , c ∈ R i.e.,if a ≤ b and b ≤ c then a ≤ c , which is true for all the values of a , b , c ∈ R . For example: if a = 2 , b = 3 and c = 4 Here, a < b < c So, the given relation is transitive. Hence, the given relation R is reflexive and transitive but not symmetric.

Let N be the set of natural numbers and a relation R on N be defined by R = x , y ∈ N × N : x 3 - 3 x 2 y - x y 2 + 3 y 3 = 0 . Then the relation R is

reflexive but neither symmetric nor transitive

symmetric but neither reflexive nor transitive

reflexive and symmetric, but not transitive

Let A = { 1 , 2 , 3 , 4 , 5 } and R be a relation defined by R = { ( x , y ) : x , y ∈ A , x + y = 5 } . Then, R is

neither reflexive nor transitive

reflexive and symmetric but not transitive

neither reflexive nor symmetric but transitive

HARD

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Let $T$ be a set of $n$ elements. It is defined that

a relation $R$ on $T$ is symmetric if $(a, b) \in R$ then $(b, a) \in R$ for $a, b \in T$ ;

a relation $R$ on $T$ is antisymmetric if $(a, b), (b, a) \in R$ then $a = b$ for $a, b \in T$ .

Let $S$ and $A$ be the set of all symmetric and antisymmetric relations on $T$ respectively.

Then

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Important Questions on Relations and Functions

HARD

Mathematics>Algebra>Relations and Functions>Types of Relations

Let

R

and

S

be any two equivalence relations on a set

X

. Then which of the following is incorrect statement

MEDIUM

Mathematics>Algebra>Relations and Functions>Types of Relations

If a relation

R

on the set

\{1, 2, 3\}

be defined by

R = \{(1, 1)\}

, then

R

EASY

Mathematics>Algebra>Relations and Functions>Types of Relations

Let

ρ_{1}

and

ρ_{2}

be two equivalence relations defined on a non-void set

S

. Then

EASY

Mathematics>Algebra>Relations and Functions>Types of Relations

Let

R = {(P, Q) | P

and

Q

are at the same distance from the origin

}

be a relation, then the equivalence class of

(1, - 1)

is the set

EASY

Mathematics>Algebra>Relations and Functions>Types of Relations

Consider the non-empty set consisting of children in a family and a relation

R

defined as

a R b

a

is brother of

b

. Then

R

EASY

Mathematics>Algebra>Relations and Functions>Types of Relations

Let $R$ be the real line. Consider the following subsets of the plane $R \times R$

$S = {(x, y) ∣ y = x + 1, 0 < x < 2}$

$T = {(x, y) ∣ x - y$ is an integer $} .$ Which one of the following is true?

MEDIUM

Mathematics>Algebra>Relations and Functions>Types of Relations

Let the relation $\rho$ be defined on

R

a ρ b

1 + a b > 0 .

Then

MEDIUM

Mathematics>Algebra>Relations and Functions>Types of Relations

Let

S = {1, 2, 3, \dots, 24}

. Define a relation

~

S

x ~ y

if the product of the digits in

x

is same as that of the digits of

y

(Note that if

x

is a single digit number then the product of the digits in will be considered to be

x

). Then the number of equivalence classes for this equivalence relation is

HARD

Mathematics>Algebra>Relations and Functions>Types of Relations

Let Z denote the set of all integers. If a relation R is defined on Z as follows:
( x, y)

\in

R if and only if x is multiple of y, then R is

HARD

Mathematics>Algebra>Relations and Functions>Types of Relations

Let

R_{1}

and

R_{2}

be two relations defined as follows :

$R_{1} = \{(a, b) \in R^{2} : a^{2} + b^{2} \in Q\}$ and $R_{2} = \{(a, b) \in R^{2} : a^{2} + b^{2} \notin Q\}$ , where $Q$ is the set of all rational numbers, then

HARD

Mathematics>Algebra>Relations and Functions>Types of Relations

Define a relation $R$ over a class of $n \times n$ real matrices $A$ and $B$ as " $A R B$ iff there exists a non-singular matrix $P$ such that $P A P^{- 1} = B$ ". Then which of the following is true ?

MEDIUM

Mathematics>Algebra>Relations and Functions>Types of Relations

Let

R

be the real line. Consider the following subsets of the plane

R \times R

S = {(x, y) : y = x + 1

and

0 < x < 2}

T = {(x, y) : x - y

is an integer}

Which of the following is true?

EASY

Mathematics>Algebra>Relations and Functions>Types of Relations

Relation

S = {(1, 2), (2, 1), (2, 3)}

is defined on the set

{1, 2, 3}

MEDIUM

Mathematics>Algebra>Relations and Functions>Types of Relations

Let

A = \{2, 3, 4, 5, \dots ., 16, 17, 18\} .

Let

\approx

be the equivalence relation on

A \times A

cartesian product of

A

and

A

, defined by

(a, b) \approx (c, d)

a d = b c,

then the number of ordered pairs of the equivalence class of

(3, 2)

MEDIUM

Mathematics>Algebra>Relations and Functions>Types of Relations

The relation $S = {(3, 3), (4, 4)}$ on the set $A = \{3, 4, 5\}$ is

MEDIUM

Mathematics>Algebra>Relations and Functions>Types of Relations

Show that the relation

R

R

is defined as

R = {(a, b) : a \leq b}

, is reflexive and transitive but not symmetric.

MEDIUM

Mathematics>Algebra>Relations and Functions>Types of Relations

Let

N

be the set of natural numbers and a relation

R

N

be defined by

R = \{(x, y) \in N \times N : x^{3} - 3 x^{2} y - x y^{2} + 3 y^{3} = 0\}

. Then the relation

R

EASY

Mathematics>Algebra>Relations and Functions>Types of Relations

Let

A = {1, 2, 3, 4, 5}

and

R

be a relation defined by

R = {(x, y) : x, y \in A, x + y = 5}

. Then,

R

EASY

Mathematics>Algebra>Relations and Functions>Types of Relations

Let

P

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

P = \{(a, b) : \sec^{2} a - \tan^{2} b = 1\}

. Then,

P

MEDIUM

Mathematics>Algebra>Relations and Functions>Types of Relations

Let

R

be a relation on the set of all natural numbers given by

a R b \Leftrightarrow a

divides

b^{2} .

Which of the following properties does

R

satisfy?

I .

Reflexivity

I I .

Symmetry

I I I .

Transitivity

Let T be a set of n elements. It is defined that a relation R on T is symmetric if a,b∈R then b,a∈R for a,b∈T; a relation R on T is antisymmetric if a,b,b,a∈R then a=b for a,b∈T. Let S and A be the set of all symmetric and antisymmetric relations on T respectively. Then

Important Questions on Relations and Functions

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