RD SHARMA Solutions for Class 12-science Maths Chapter 1 - Relations

Page / Exercise

Chapter 1 - Relations Exercise Ex. 1.1

Question 1(i)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Solution 1(i)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Question 1(ii)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Solution 1(ii)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Question 1(iii)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Solution 1(iii)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Question 1(iv)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Solution 1(iv)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Question 2
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations
Solution 2
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations
Question 3(i)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations
Solution 3(i)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations
Question 3(ii)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations
Solution 3(ii)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations
Question 3(iii)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations
Solution 3(iii)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations
Question 4
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations
Solution 4
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations
Question 5(i)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations
Solution 5(i)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations
Question 5(ii)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Solution 5(ii)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Question 5(iii)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations
Solution 5(iii)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations
Question 6

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Solution 6

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Question 7

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Solution 7

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Question 8
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations
Solution 8
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations
Question 9
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations
Solution 9
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations
Question 10
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations
Solution 10
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations
Question 11
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations
Solution 11
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations
Question 12
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations
Solution 12
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations
Question 13

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Solution 13

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Question 14

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 14

 

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

 

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

 

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Question 15

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Solution 15

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Question 16

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Solution 16

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Question 17

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 17

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

 

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 18

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 18

A relation R in A is said to be reflexive if aRa for all aA, R is symmetric if aRb bRa, for all a, b A and it is said to be transitive if aRb and bRc aRc 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, 9)ϵR 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
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations
Solution 1
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations
Question 2
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations
Solution 2
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations
Question 3
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations
Solution 3
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations
Question 4
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations
Solution 4
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations
Question 5
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations
Solution 5
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations
Question 6
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations
Solution 6
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations
Question 7
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations
Solution 7
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations
Question 8

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Solution 8

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

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
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations
Solution 9
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations
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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations


Question 11
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations
Solution 11
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations
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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations


Question 13
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations
Solution 13
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations
Question 14

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Solution 14

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - RelationsRd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Question 15
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations
Solution 15
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations
Question 16
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations
Solution 16
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

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,

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Solution 1

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations


Question 2

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

 

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Solution 2

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Question 3

R is a relation on the set Z of integers and it is given by Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

(a) reflexive and transitive

(b) reflexive and symmetric

(c) symmetric and transitive

(d) an equivalence relation

Solution 3

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Question 4

The relation R defined on the set A = {1, 2, 3, 4, 5} by Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations 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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Question 5

Let R be the relation over the set of all straight lines in a plane such that l1 R l2 Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations  l1Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relationsl2. Then, R is

(a) symmetric

(b) reflexive

(c) transitive 

(d) an equivalence relation

Solution 5

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Question 7

Let A = {2, 3, 4, 5, ...., 17, 18}. Let Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relationsbe 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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Question 9

The relation 'R' in N × N such that (a, b) R (c, d) Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations 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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Question 11

A relation R is defined from {2, 3, 4, 5} to {3, 6, 7, 10} by : x R y Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations 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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Question 12

A relation Φ from C to R is defined by  Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations Which one is correct ?

 Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Solution 12

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Question 17

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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Solution 17

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

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 Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations 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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Question 25

S is relation over the set R of all real numbers and it is given by (a, b) ε S Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations ab ≥ 0. Then, S is

(a) symmetric and transitive only

(b) reflexive and symmetric only

(c) antisymmetric relation

(d) an equivalence relation

Solution 25

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Question 29

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

 

(a) 1

(b) 2

(c) 3

(d) 5

Solution 29

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Question 30

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 30

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Question 31

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 31

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Question 32

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 32

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Question 33

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 33

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Question 34

For real numbers x and y, define x R y if x - y + Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations is an irrational number. Then the relation R is

(a) reflexive

(b) symmetric

(c) transitive

(d) none of these

Solution 34

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Chapter 1 - Relations Exercise Ex. 1VSAQ

Question 1

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Solution 1

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Question 2

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Solution 2

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Question 3

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Solution 3

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Question 4

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Solution 4

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Question 5

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Solution 5

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Question 6

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Solution 6

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Question 7

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Solution 7

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Question 8

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Solution 8

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Question 9

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Solution 9

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Question 10

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Solution 10

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Question 11

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Solution 11

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Question 12

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Solution 12

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Question 13

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Solution 13

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Question 14

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Solution 14

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Question 15

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Solution 15

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Question 16

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Solution 16

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Question 17

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

Solution 17

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Relations

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 aA, R is symmetric if aRb bRa, for all a, b A and it is said to be transitive if aRb and bRc aRc 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 aA, R is symmetric if aRb bRa, for all a, b A and it is said to be transitive if aRb and bRc aRc 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- b2 < 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 aN

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 aA, R is symmetric if aRb bRa, for all a, b A and it is said to be transitive if aRb and bRc aRc 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.