You cannot copy content of this page

# RD Sharma Solution CLass 12 Mathematics Chapter 1 Relations

## Chapter 1 – Relations Exercise Ex. 1.1

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5
Solution 5
Question 6
Solution 6
Question 7
Solution 7
Question 8
Solution 8
Question 9
Solution 9
Question 10
Solution 10
Question 11

Solution 11

Question 12
Solution 12
Question 13

Solution 13

Question 14

Solution 14

Question 15
Solution 15
Question 16
Solution 16
Question 17
Solution 17
Question 18
Solution 18
Question 19
Solution 19
Question 20

Solution 20

Question 21

Give an Example of a relation which is

1. Reflexive and symmetric but not transitive
2. Reflexive and transitive but not symmetric
3. Symmetric and transitive but not reflexive
4. Symmetric but neither reflexive nor transitive
5. Transitive but neither reflexive nor symmetric
Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Let A = {a, b, c} and the relation R be defined
on A as follows:

R = {(a,
a), (b, c), (a, b)}.

Then, write minimum number of ordered pairs to
be added in R to make it reflexive and transitive.

[ NCERT EXEMPLAR ]

Solution 24

A relation R in A is said to be reflexive if aRa for all a∈A

R is said to be transitive if aRb and bRc aRc

for all a, b, c ∈ A.

Hence for R to be reflexive (b, b) and (c, c)
must be there in the set R.

Also for R to be transitive (a, c) must be in R
because (a, b)

R and (b, c) ∈
R so (a, c)
must be in R.

So at least 3 ordered pairs must be added for R
to be reflexive and transitive.

Question 25

Each of the following defines a relation on N:

• x > y,  x, y ϵ N

• x + y = 10,  x, y ϵ N

• xy is square of an integer,  x, y
ϵ N

• x + 4y = 10,  x, y ϵ N

Determine which of the above relations are
reflexive, symmetric and transitive.

[ NCERT EXEMPLAR ]

Solution 25

A relation R in A is said to be reflexive if aRa for all a∈A, R is symmetric
if aRb
bRa, for all a, b ∈ A and it is said to be transitive if aRb and bRcaRc for all
a, b, c ∈ A.

• x > y,  x, y ϵ N

(x, y)
ϵ
{(2, 1), (3, 1)…….(3, 2), (4, 2)….}

This is not reflexive as (1, 1), (2, 2)….are
absent.

This is not symmetric as (2,1)
is present but (1,2) is absent.

This is transitive as (3, 2) ϵ
R and (2,1) ϵ R also (3,1) ϵ
R ,similarly
this property satisfies all cases.

• x + y = 10,  x, y ϵ N

(x, y)ϵ {(1, 9), (9, 1), (2, 8), (8, 2), (3, 7),
(7, 3), (4, 6), (6, 4), (5, 5)}

This is
not reflexive as (1, 1),(2, 2)….. are absent.

This
only follows the condition of symmetric set as  (1, 9R
also (9, 1)
ϵR similarly other cases are also satisfy the condition.

This is not transitive because {(1, 9),(9, 1)}ϵR but (1, 1) is absent.

• xy is square of an integer, x, y ϵ N

(x, y)
ϵ
{(1, 1), (2, 2),
(4, 1), (1, 4), (3, 3), (9, 1),  (1, 9), (4, 4), (2, 8), (8, 2), (16,
1), (1, 16)………..}

This is
reflexive as (1,1),(2,2)….. are present.

This is
also symmetric because if aRb
bRa, for all
a,bϵN.

This is
transitive also because if aRb and bRc
aRc for all a, b, c ϵ N.

• x + 4y = 10, x, y ϵ N

(x, y) ϵ {(6, 1), (2, 2)}

This is
not reflexive as (1, 1), (2, 2)…..are absent.

This is
not symmetric because (6,1)
ϵ
R but (1,6) is
absent.

This is not transitive as there are only two
elements in the set having no element common.

## Chapter 1 – Relations Exercise Ex. 1.2

Question 1
Solution 1
Question 2
Solution 2
Question 3
Solution 3
Question 4
Solution 4
Question 5
Solution 5
Question 6
Solution 6
Question 7
Solution 7
Question 8

Solution 8

Also we need to find the set of all elements related to 1.

Since the relation is given by, R={(a,b):a=b}, and 1 is an element of A,

R={(1,1):1=1}

Thus, the set of all elements related to 1 is 1.

Question 9
Solution 9
Question 10

Show that the relation R, defined on the set A of all polygons as

R = {(p1, p2) : p1 and p2 have same number of sides},is an equivalence relation.

What is the set of all elements in A related to the right angle triangle T with sides 3, 4, and 5?

Solution 10

Question 11
Solution 11
Question 12

Let R be the relation defined on the set A = {1, 2, 3, 4, 5, 6, 7} by R = {(a, b) : both a and b are either odd or even}.

Show that R is an equivalence relation. Further, show that all the elements of the subset {1, 3, 5, 7} are related to each other and all the elements of the subset {2, 4, 6} are related to each other but no element of the subset {1, 3, 5, 7} is related to any element of the subset {2, 4, 6}.

Solution 12

Question 13
Solution 13
Question 14

Solution 14

Question 15
Solution 15
Question 16
Solution 16

## Chapter 1 – Relations Exercise Ex. 1VSAQ

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Let
R = {(a, a3): a is a prime
number less than 5} be a relation. Find the range of R.

Solution 18

R
= {(2,8),(3, 27)}

The range set of R is {8, 27}.

Question 19

Let R be the equivalence relation on the set Z
of integers given by R = {(a, b): 2 divides a – b}.

Write the equivalence class [0].

[ NCERT EXEMPLAR ]

Solution 19

a, b ∈ Z and R is given by R={(a, b): 2
divides a-b}.

The equivalence classes can be taken as [0],
[1].

Note that, for 0 i ≤ 1, [i] = {2n
+ i :
n ∈ Z}.

So equivalence class [0] = {2n : n ∈ Z}

It is clear that all the elements of
equivalence class [0] are even.

Hence, equivalence class [0]={0,
± 2, ± 4, ± 6,…}

Question 20

For the set A = {1, 2, 3}, definite a relation
R on the set A as follows:

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

Write the ordered pairs to be added to R to
make the smallest equivalence relation.

Solution 20

A relation R in A is said to be reflexive if aRa for all a∈A, R is symmetric
if aRb
bRa, for all a, b ∈ A and it is said to be transitive if aRb and bRcaRc for all
a, b, c ∈ A. Any relation which is reflexive, symmetric and transitive is
called an equivalence relation.

Given R is reflexive and transitive but not
symmetric hence to make symmetric (3,1) added to R
because (1,3) is already there. Hence (3, 1) is the single ordered pair which
needs to be added to R to make it the smallest equivalence relation.

Question 21

Let A = {0, 1, 2, 3} and R be a relation on A
defined as

R = {(0, 0), (0, 1), (0, 3), (1, 0), (1, 1),
(2, 2), (3, 0), (3, 3)}

Is R reflexive? Symmetric? Transitive?

Solution 21

A relation R in A is said to be reflexive if aRa for all a∈A, R is symmetric
if aRb
bRa, for all a, b ∈ A and it is said to be transitive if aRb and bRcaRc for all
a, b, c ∈ A

R is reflexive because {(0, 0),(1, 1),(2, 2),(3, 3)}∈R .

R is symmetric, because {(0,1),(0,3)}∈R and {(1,0),(3,0)} ∈
R .

but R is not transitive since for (1, 0) ∈ R and(0, 3) ∈ R whereas (1, 3) ∉ R.

Question 22

Let the relation R be defined on the set

A = {1, 2, 3, 4, 5} by R = {(a, b) : ⃒a2– b2⃒ < 8}.

Write R as a set of ordered pairs.

Solution 22

A = {1, 2, 3, 4, 5} and R = {(a, b) :|a2
– b2 | < 8}

If a = 1 then from R we get b = 1, 2

If a = 2 then from R we get b = 1, 2, 3

If a = 3 then from R we get b = 2, 3, 4

If a = 4 then from R we get b = 3, 4

If a = 5 then from R we get b = 5

Hence R={(1,1),(1,2),(2,1),(2,2),(2,3),(3,2),(3,3),(3,4),(4,3), (4,4),(5,5)}

Question 23

Let the relation R be define on N by aRb iff

2a + 3b
= 30. Then write R as a set of ordered pairs.

Solution 23

If a, b ∈ N then b must be an even integer so that a∈N

Hence only possible values for b are 2,4,6,8.

if b=2 , it gives a=12

if b=4 , it gives a=9

if b=6 , it gives a=6

if b=8 , it gives a=3

hence (a,b)ϵ{(3,8),(6,6),(9,4),(12,2)}

Question 24

Write the smallest equivalence relation on the
set A = {1, 2, 3}.

Solution 24

A relation R in A is said to be reflexive if aRa for all a∈A, R is symmetric
if aRb
bRa, for all a, b ∈ A and it is said to be transitive if aRb and bRcaRc for all
a, b, c ∈ A. Any relation which is reflexive, symmetric and transitive is
called an equivalence relation.

Hence smallest possible equivalence relation is
{(1,1),(2,2),(3,3)}.

This relation satisfies all the conditions for
an equivalence relation.

## Chapter 1 – Relations Exercise MCQ

Question 1

Let R be a relation on the set N given by R = {(a, b) : a = b – 2, b > 6}. Then,

Solution 1

Question 2

which of the following is not an equivalence relation on Z?

Solution 2

Question 3

R is a relation on the set Z of integers and it is given by

(a) reflexive and transitive

(b) reflexive and symmetric

(c) symmetric and transitive

(d) an equivalence relation

Solution 3

Question 4

The relation R defined on the set A = {1, 2, 3, 4, 5} by  is given by

(a) {(1, 1), (2, 1), (3, 1), (4, 1), (2, 3)}

(b) {(2, 2), (3, 2), (4, 2), (2, 4)}

(c) {(3, 3), (4, 3), (5, 4), (3, 4)}

(d) none of these

Solution 4

Question 5

Let R be the relation over the set of all straight lines in a plane such that l1 R l2   l1l2. Then, R is

(a) symmetric

(b) reflexive

(c) transitive

(d) an equivalence relation

Solution 5

Question 6

if A = {a, b, c}, then the relation R = {(b, c)} on A is

(a) reflexive only

(b) symmetric only

(c) transitive

(d) reflexive and transitive only

Solution 6

Question 7

Let A = {2, 3, 4, 5, …., 17, 18}. Let be the equivalence relation on A × A, cartesian product of A with itself, defined by (a, b) (c, d) iff ad = bc. Then, the number of ordered pairs of the equivalence class of (3, 2) is

(a) 4

(b) 5

(c) 6

(d) 7

Solution 7

Question 8

Let A = {1, 2, 3}. Then, the number of relations containing (1, 2) and (1, 3) which are reflexive and symmetric but not transitive is

(a) 1

(b) 2

(c) 3

(d) 4

Solution 8

Question 9

The relation ‘R’ in N × N such that (a, b) R (c, d)  a + d = b + c is

(a) reflexive but not symmetric

(b) reflexive and transitive but not symmetric

(c) an equivalence relation

(d) none of these

Solution 9

Question 10

If A = {1, 2, 3}, B = {1, 4, 6, 9} and R is a relation from A to B defined by ‘x is greater than y’. The range of R is

(a) {1, 4, 6, 9}

(b) {4, 6, 9}

(c) {1}

(d) none of these

Solution 10

Question 11

A relation R is defined from {2, 3, 4, 5} to {3, 6, 7, 10} by : x R y  x is relatively prime to y. Then, domain of R is

(a) {2, 3, 5}

(b) {3, 5]

(c) {2, 3, 4}

(d) {2, 3, 4, 5}

Solution 11

Question 12

A relation Φ from C to R is defined by   Which one is correct ?

Solution 12

Question 13

Let R be a relation on N defined by x + 2y = 8. The domain of R is

(a) {2, 4, 8}

(b) {2, 4, 6, 8}

(c) {2, 4, 6}

(d) {1, 2, 3, 4}

Solution 13

Question 14

R is a relation from {11, 12, 13} to {8, 10, 12} defined by y = x – 3. Then, R is

(a) {(8, 11), (10, 13)}

(b) {(11, 8), (13, 10)}

(c) {(10, 13), (8, 11), (8, 10)}

(d) none of these

Solution 14

Question 15

Let R = {(a, a), (b, b), (c, c), (a, b)} be a relation on set A = {a, b, c}. Then, R is

(a) identity relation

(b) reflexive

(c) symmetric

(d) equivalence

Solution 15

Question 16

Let A = {1, 2, 3} and R = {(1, 2), (2, 3), (1, 3)} be a relation on A. Then, R is

(a) neither reflexive nor transitive

(b) neither symmetric nor transitive

(c) transitive

(d) none of these

Solution 16

Question 17

If R is the largest equivalence relation on a set A and S is any relation on A, then

Solution 17

Question 18

If R is a relation on the set A = {1, 2, 3, 4, 5, 6, 7, 8, 9} given by x R y  y = 3x, then R =

(a) {(3, 1), (6, 2), (8, 2), (9, 3)}

(b) {(3, 1), (6, 2), (9, 3)}

(c) {(3, 1), (2, 6), (3, 9)}

(d) none of these

Solution 18

Question 19

If R is a relation on the set A = {1, 2, 3} given by R = {(1, 1), (2, 2), (3, 3)} then R is

(a) reflexive

(b) symmetric

(c) transitive

(d) all the three options

Solution 19

Question 20

If A = {a, b, c, d} then a relation R = {(a, b), (b, a), (a, a)} on A is

(a) symmetric and transitive only

(b) reflexive and transitive only

(c) symmetric only

(d) transitive only

Solution 20

Question 21

If A = {1, 2, 3}, then a relation R = {(2, 3)} on A is

(a) symmetric and transitive only

(b) stmmetric only

(c) transitive only

(d) none of these

Solution 21

Question 22

Let R be the relation on the set A = {1, 2, 3, 4} given by

R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}. Then,

(a) R is reflexive and symmetric but not transitive

(b) R is reflexive and transitive but not not symmetric

(c) R is symmetric and transitive but not reflexive

(d) R is an equivalence relation

Solution 22

Question 23

Let A = {1, 2, 3}. Then, the number of equivalence relation containing (1, 2) is

(a) 1

(b) 2

(c) 3

(d) 4

Solution 23

Question 24

The relation R = {(1, 1), (2, 2), (3, 3)} on the set {1, 2, 3} is

(a) symmetric only

(b) reflexive only

(c) an equivalence relation

(d) transitive only

Solution 24

Question 25

S is relation over the set R of all real numbers and it is given by (a, b) ε S  ab ≥ 0. Then, S is

(a) symmetric and transitive only

(b) reflexive and symmetric only

(c) antisymmetric relation

(d) an equivalence relation

Solution 25

Question 26

In the set Z of all integers, which of the following relation R is not an equivalence relation?

(a) x R y : if x ≤ y

(b) x R y : if x = y

(c) x R y : if x – y is an even integer

(d) x R y : if x ≡ y (mod 3)

Solution 26

Question 27

Let A = {1, 2, 3} and consider the relation R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)}. Then, R is

(a) reflexive but not symmetric

(b) reflexive but not transitive

(c) symmetric and transitive

(d) neither symmetric nor transitive

Solution 27

Question 28

The maximum number of equivalence relations on the set A = {1, 2, 3} is

(a) 1

(b) 2

(c) 3

(d) 5

Solution 28

Question 29

let R be a relation on the set N of a natural numbers defined by n R m iff n divides m. Then, R is

(a) Reflexive and symmetric

(b) Transitive and symmetric

(c) Equivalence

(d) Reflexive, transitive but not symmetric

Solution 29

Question 30

Let L denote the set of all straight lines in a plane. Let a relation R be defined by l R m iff l is perpendicular to m for all l, m ε L. Then, R is

(a) reflexive

(b) symmetric

(c) transitive

(d) one of these

Solution 30

Question 31

let T be the set of all triangles in the Euclidean plane, and let a relation R on T be defined as a R b if a is congruent to b for all a, b ε T. Then, R is

(a) Reflexive but not symmetric

(b) transitive but not symmetric

(c) equivalence

(d) none of these

Solution 31

Question 32

Consider a non-empty set consisting of children in a family and a relation R defined as a R b if a is brother of b. Then, R is

(a) symmetric but not transitive

(b) transitive but not symmetric

(c) neither symmetric nor transitive

(d) both symmetric transitive

Solution 32

Question 33

For real numbers x and y, define x R y if x – y +  is an irrational number. Then the relation R is

(a) reflexive

(b) symmetric

(c) transitive

(d) none of these

Solution 33

error: Content is protected !!