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

Let the relation ρ be defined on R by a ρ b holds if and only if a - b is zero or irrational, then

ρ is reflexive and symmetric but is not transitive

ρ is reflexive and transitive but is not symmetric

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 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

Earn 100

Let a relation $R$ is defined on a set of an involutory square matrix of order $2 .$ If $A, B$ are two such matrices and relation $R$ is defined as $(A, B) \in R \Rightarrow (A B)^{⊤} = A^{⊤} B^{⊤}$ , then relation $R$ is ______

50% studentsanswered this correctly

Important Questions on Set Theory and Relations

MEDIUM

Mathematics>Algebra>Set Theory and Relations>Types of Relations

R

, a relation

ρ

is defined by

x ρ y

if and only if

x - y

is zero or irrational. Then,

MEDIUM

Mathematics>Algebra>Set Theory and Relations>Types of Relations

If a relation

R

on the set

\{1, 2, 3\}

be defined by

R = \{(1, 1)\}

, then

R

MEDIUM

Mathematics>Algebra>Set Theory and Relations>Types of Relations

On the set

R

of real numbers, the relation

ρ

is defined by

x ρ y, (x, y) \in R

EASY

Mathematics>Algebra>Set Theory and Relations>Types of Relations

Let

ρ_{1}

and

ρ_{2}

be two equivalence relations defined on a non-void set

S

. Then

EASY

Mathematics>Algebra>Set Theory and Relations>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

MEDIUM

Mathematics>Algebra>Set Theory and Relations>Types of Relations

Let

ρ

be a relation defined on

N,

the set of natural numbers, as

ρ = {(x, y) \in N \times N : 2 x + y = 41} .

Then

MEDIUM

Mathematics>Algebra>Set Theory and Relations>Types of Relations

Let the relation $\rho$ be defined on

R

a ρ b

1 + a b > 0 .

Then

EASY

Mathematics>Algebra>Set Theory and Relations>Types of Relations

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

then which one of the following is reflexive?

HARD

Mathematics>Algebra>Set Theory and Relations>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>Set Theory and Relations>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>Set Theory and Relations>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>Set Theory and Relations>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?

MEDIUM

Mathematics>Algebra>Set Theory and Relations>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>Set Theory and Relations>Types of Relations

Let the relation

ρ

be defined on

R

a ρ b

holds if and only if

a - b

is zero or irrational, then

EASY

Mathematics>Algebra>Set Theory and Relations>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?

HARD

Mathematics>Algebra>Set Theory and Relations>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 ?

EASY

Mathematics>Algebra>Set Theory and Relations>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>Set Theory and Relations>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>Set Theory and Relations>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>Set Theory and Relations>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 a relation R is defined on a set of an involutory square matrix of order 2. If A, B are two such matrices and relation R is defined as (A,B)∈R⇒(AB)⊤=A⊤B⊤, then relation R is ______

Important Questions on Set Theory and Relations

Learn and Prepare for any exam you want !

Let a relation $R$ is defined on a set of an involutory square matrix of order $2 .$ If $A, B$ are two such matrices and relation $R$ is defined as $(A, B) \in R \Rightarrow (A B)^{⊤} = A^{⊤} B^{⊤}$ , then relation $R$ is ______