Check the relation on the set A = 1 , 2 , 3 , 4 , 5 , 6 by R = ( a , b ) ∈ R : | a - b | ≥ 0 is the universal relation.

We now that, A relation R in a set A is said to be universal relation, if each element of A is related to every element of A . i.e, R = A × A From the given, The relation on the set A = 1 , 2 , 3 , 4 , 5 , 6 by R = ( a , b ) ∈ R : | a - b | ≥ 0 is the universal relation. We observe that | a - b | ≥ 0 for all a , b ∈ A ⇒ ( a , b ) ∈ R for all ( a , b ) ∈ A X A Each element of set A is related to every element of set A . ⇒ R = A X A ⇒ R is a universal relation on set A .

Check the relation on the set A = 1 , 2 , 3 , 4 , 5 , 6 by R = ( a , b ) ∈ R : | a - b | ≥ 0 is the universal relation.

We now that, A relation R in a set A is said to be universal relation, if each element of A is related to every element of A . i.e, R = A × A From the given, The relation on the set A = 1 , 2 , 3 , 4 , 5 , 6 by R = ( a , b ) ∈ R : | a - b | ≥ 0 is the universal relation. We observe that | a - b | ≥ 0 for all a , b ∈ A ⇒ ( a , b ) ∈ R for all ( a , b ) ∈ A X A Each element of set A is related to every element of set A . ⇒ R = A X A ⇒ R is a universal relation on set A .

Let N denote the set of all natural numbers. Define two binary relations on N as R 1 = x , y ∈ N × N : 2 x + y = 10 and R 2 = x , y ∈ N × N : x + 2 y = 10 . Then

both R 1 and R 2 are transitive relations

both R 1 and R 2 are symmetric relations

Check whether the relation R defined in the set 1 , 2 , 3 , 4 , 5 , 6 as R = a , b : b = a + 1 is reflexive, symmetric or transitive.

Suppose A = 1 , 2 , 3 , 4 , 5 , 6 . A relation R is defined on set A as: R = { ( a , b ) : b = a + 1 } ∴ R = { ( 1, 2 ) , 2 , 3 , 3 , 4 , 4 , 5 , 5 , 6 } Reflexive Property : A relation R in a set A is said to be reflexive if x , x ∈ R , for every x ∈ A Now, 1 , 1 , 2 , 2 , 3 , 3 , 4 , 4 , 5 , 5 , 6 , 6 ∉ R ∴ R is not reflexive. Symmetric Property : A relation R in a set A is said to be symmetric if x , y ∈ R implies that y , x ∈ R for all x , y ∈ A . Now it can be observed that 1 , 2 ∈ R , but 2 , 1 ∉ R . Hence , R is not symmetric. Transitive Property : A relation R in a set A is said to be transitive if x , y ∈ R and y , z ∈ R implies that x , z ∈ R for all x , y , z ∈ A . Now, 1 , 2 , 2 , 3 ∈ R but, ( 1 , 3 ) ∉ R ∴ R is not transitive. Hence, R is neither reflexive, nor symmetric, nor transitive.

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 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

The relation R defined in the set 1 , 2 , 3 , 4 , 5 , 6 as R = a , b : b = a + 1 is

neither reflexive nor 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 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

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

MEDIUM

Earn 100

Check the relation on the set $A = \{1, 2, 3, 4, 5, 6\}$ by $R = (a, b) \in R : | a - b | \geq 0$ is the universal relation.

Important Questions on Sets,Relations and Functions

MEDIUM

Mathematics>Algebra>Sets,Relations and Functions>Relations and its Types

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

EASY

Mathematics>Algebra>Sets,Relations and Functions>Relations and its Types

Let

N

denote the set of all natural numbers. Define two binary relations on

N

R_{1} = \{(x, y) \in N \times N : 2 x + y = 10\}

and

R_{2} = \{(x, y) \in N \times N : x + 2 y = 10\}

. Then

EASY

Mathematics>Algebra>Sets,Relations and Functions>Relations and its Types

Check whether the relation

R

defined in the set

\{1, 2, 3, 4, 5, 6\}

R = \{(a, b) : b = a + 1\}

is reflexive, symmetric or transitive.

HARD

Mathematics>Algebra>Sets,Relations and Functions>Relations and its Types

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

MEDIUM

Mathematics>Algebra>Sets,Relations and Functions>Relations and its Types

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

EASY

Mathematics>Algebra>Sets,Relations and Functions>Relations and its Types

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

EASY

Mathematics>Algebra>Sets,Relations and Functions>Relations and its Types

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>Sets,Relations and Functions>Relations and its Types

Let the relation $\rho$ be defined on

R

a ρ b

1 + a b > 0 .

Then

MEDIUM

Mathematics>Algebra>Sets,Relations and Functions>Relations and its Types

A set

P = {7, 21, 28, 42, 14, 0}

is given. A relation

R

holds on elements

a

and

b

of set

P

if either

a

is a factor of

b

b

is a factor of

a .

The relation

R

on the set

P

MEDIUM

Mathematics>Algebra>Sets,Relations and Functions>Relations and its Types

The relation

R

defined in the set

\{1, 2, 3, 4, 5, 6\}

R = \{(a, b) : b = a + 1\}

HARD

Mathematics>Algebra>Sets,Relations and Functions>Relations and its Types

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

MEDIUM

Mathematics>Algebra>Sets,Relations and Functions>Relations and its Types

R

, a relation

ρ

is defined by

x ρ y

if and only if

x - y

is zero or irrational. Then,

EASY

Mathematics>Algebra>Sets,Relations and Functions>Relations and its Types

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

MEDIUM

Mathematics>Algebra>Sets,Relations and Functions>Relations and its Types

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>Sets,Relations and Functions>Relations and its Types

If a relation

R

on the set

\{1, 2, 3\}

be defined by

R = \{(1, 1)\}

, then

R

HARD

Mathematics>Algebra>Sets,Relations and Functions>Relations and its Types

Let

R

and

S

be any two equivalence relations on a set

X

. Then which of the following is incorrect statement

EASY

Mathematics>Algebra>Sets,Relations and Functions>Relations and its Types

Let

ρ_{1}

and

ρ_{2}

be two equivalence relations defined on a non-void set

S

. Then

EASY

Mathematics>Algebra>Sets,Relations and Functions>Relations and its Types

In the set

ℕ

of natural numbers, the relation

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

MEDIUM

Mathematics>Algebra>Sets,Relations and Functions>Relations and its Types

Let the relation

ρ

be defined on

R

a ρ b

holds if and only if

a - b

is zero or irrational, then

MEDIUM

Mathematics>Algebra>Sets,Relations and Functions>Relations and its Types

On the set

R

of real numbers, the relation

ρ

is defined by

x ρ y, (x, y) \in R

Check the relation on the set A=1,2,3,4,5,6 by R=(a,b) ∈R:|a-b|≥0 is the universal relation.

Important Questions on Sets,Relations and Functions

Learn and Prepare for any exam you want !

Check the relation on the set $A = \{1, 2, 3, 4, 5, 6\}$ by $R = (a, b) \in R : | a - b | \geq 0$ is the universal relation.