Chapter 1 : Relations - Rd Sharma Solutions for Class 12-science Maths CBSE

Your CBSE Class 12 syllabus for Maths consists of topics such as linear programming, vector quantities, determinants, etc. which lay the foundation for further education in science, engineering, management, etc. Studying differential equations will be useful for exploring subjects such as Physics, Biology, Chemistry, etc. where your knowledge can be applied for scientific investigations.

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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 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 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 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
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 MCQ

Question 1

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

begin mathsize 12px style open parentheses straight a close parentheses space open parentheses 2 comma space 4 close parentheses space element of straight R
open parentheses straight b close parentheses space open parentheses 3 comma space 8 close parentheses element of space straight R
open parentheses straight c close parentheses space open parentheses 6 comma space 8 close parentheses space element of space straight R
open parentheses straight d close parentheses space open parentheses 8 comma space 7 close parentheses space element of straight R end style

Solution 1

begin mathsize 12px style Correct space option colon space left parenthesis straight c right parenthesis
For space option space straight c space open parentheses 6 comma 8 close parentheses
straight b equals 8 greater than 6
Also space straight a equals straight b minus 2
Hence comma space answer space is space open parentheses 6 comma 8 close parentheses
end style


Question 2

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

 

Error converting from MathML to accessible text.

Solution 2

begin mathsize 12px style Correct space option space colon space left parenthesis straight c right parenthesis
As space straight a space straight R space straight b space left right double arrow straight a less than straight b
does space not space satisfy space reflexive space and space symmetric space relation.
Hence comma space answer space is space left parenthesis straight c right parenthesis. end style

Question 3

R is a relation on the set Z of integers and it is given by begin mathsize 12px style left parenthesis straight x comma space straight y right parenthesis space element of space straight R space left right double arrow open vertical bar straight x minus straight y close vertical bar less or equal than 1. end style

(a) reflexive and transitive

(b) reflexive and symmetric

(c) symmetric and transitive

(d) an equivalence relation

Solution 3

begin mathsize 12px style Correct space option colon space left parenthesis straight b right parenthesis
For space open parentheses straight x comma straight y close parentheses element of straight R space left right double arrow open vertical bar straight x minus straight y close vertical bar less or equal than 1
For space straight x equals straight y space rightwards double arrow space open vertical bar straight x minus straight y close vertical bar less or equal than 1
Also comma space
straight f open parentheses straight x comma straight y close parentheses equals straight f open parentheses straight y comma straight x close parentheses
As space mod space function space is space given.
Hence comma space function space satisfies space reflexive space and space symmetric space relation space on space straight Z.

Also space for space straight x equals straight y space semicolon space open vertical bar straight x minus straight y close vertical bar less or equal than 1 condition space is space satisfied. end style

Question 4

The relation R defined on the set A = {1, 2, 3, 4, 5} by begin mathsize 12px style straight R space equals open curly brackets open parentheses straight a comma straight b close parentheses semicolon open vertical bar straight a squared minus straight b squared close vertical bar space less than space 16 close curly brackets comma end style 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

begin mathsize 12px style Correct space option colon left parenthesis straight d right parenthesis
Given space straight A equals open curly brackets 1 comma 2 comma 3 comma 4 comma 5 close curly brackets space and space straight R equals left curly bracket open parentheses straight a comma straight b close parentheses semicolon space open vertical bar straight a squared minus straight b squared close vertical bar less than 16
As space per space given space information
open curly brackets open parentheses 1 comma 1 close parentheses comma open parentheses 1 comma 2 close parentheses comma open parentheses 1 comma 3 close parentheses comma open parentheses 1 comma 4 close parentheses comma open parentheses 1 comma 5 close parentheses comma open parentheses 2 comma 1 close parentheses comma open parentheses 2 comma 2 close parentheses comma open parentheses 2 comma 3 close parentheses comma open parentheses 2 comma 4 close parentheses comma open parentheses 2 comma 5 close parentheses comma open parentheses 3 comma 1 close parentheses comma open parentheses 3 comma 2 close parentheses comma open parentheses 3 comma 3 close parentheses comma open parentheses 3 comma 4 close parentheses comma open parentheses 3 comma 5 close parentheses comma open parentheses 4 comma 1 close parentheses comma open parentheses 4 comma 2 close parentheses comma open parentheses 4 comma 3 close parentheses comma open parentheses 4 comma 4 close parentheses comma open parentheses 4 comma 5 close parentheses comma open parentheses 5 comma 1 close parentheses comma open parentheses 5 comma 2 close parentheses comma open parentheses 5 comma 3 close parentheses comma open parentheses 5 comma 4 close parentheses comma open parentheses 5 comma 5 close parentheses close curly brackets
No space such space pairs space are space in space the space option space left parenthesis straight a right parenthesis comma space left parenthesis straight b right parenthesis comma space left parenthesis straight c right parenthesis.
hence comma space the space answer space is space left parenthesis straight d right parenthesis. end style

Question 5

Let R be the relation over the set of all straight lines in a plane such that l1 R l2 begin mathsize 12px style left right double arrow end style  l1begin mathsize 12px style perpendicular end stylel2. Then, R is

(a) symmetric

(b) reflexive

(c) transitive 

(d) an equivalence relation

Solution 5

begin mathsize 12px style Correct space option colon space left parenthesis straight a right parenthesis
Given space straight R space is space the space relation space over space the space set space of space all space straight space lines space in space straight a space plane space such space that
straight l subscript 1 space end subscript Rl subscript 2 space left right double arrow l subscript 1 space end subscript perpendicular l subscript 2 space
it space is space symmetric space relation space as space we space can space say space either space or space straight l subscript 2 space end subscript perpendicular straight l subscript 1. end style

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

begin mathsize 12px style Correct space option colon left parenthesis straight c right parenthesis
Given space straight R equals open curly brackets open parentheses straight b comma straight c close parentheses close curly brackets
Only space one space element space in space straight R space it space has space transitive space relation. end style

Question 7

Let A = {2, 3, 4, 5, ...., 17, 18}. Let asymptotically equal tobe 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

begin mathsize 12px style Correct space option colon left parenthesis straight c right parenthesis
Given space that space be space the space equivalence space relation space on space straight A space cross times space straight A comma space
cartesian space product space of space straight A space with space itself comma space defined space by space left parenthesis straight a comma space straight b right parenthesis space left parenthesis straight c comma space straight d right parenthesis space iff space ad space equals space bc
To space find space the space equivalence space class space of space open parentheses 3 comma 2 close parentheses space we space will space take space element space straight a space as space multiple space of space 3
and space element space straight b space as space multiple space of space 2.
Then space the space pairs space are space
open curly brackets open parentheses 3 comma 2 close parentheses comma open parentheses 6 comma 4 close parentheses comma open parentheses 9 comma 6 close parentheses comma open parentheses 12 comma 8 close parentheses comma open parentheses 15 comma 10 close parentheses comma open parentheses 18 comma 12 close parentheses close curly brackets
Hence comma space number space of space pairs space are space 6. end style

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

begin mathsize 12px style Correct space option colon space left parenthesis straight a right parenthesis
Given space that space space straight A space equals space left curly bracket 1 comma space 2 comma space 3 right curly bracket.
To space find space space the space number space of space relations space containing space left parenthesis 1 comma space 2 right parenthesis space and space left parenthesis 1 comma space 3 right parenthesis space
then space straight R space can space be space written space as space
open curly brackets open parentheses 1 comma 2 close parentheses comma open parentheses 1 comma 3 close parentheses comma open parentheses 1 comma 1 close parentheses comma open parentheses 2 comma 2 close parentheses comma open parentheses 3 comma 3 close parentheses comma open parentheses 2 comma 1 close parentheses comma open parentheses 3 comma 1 close parentheses close curly brackets
Here comma space we space can space see space that space
left parenthesis 3 comma 1 right parenthesis space and space left parenthesis 1 comma 2 right parenthesis space rightwards double arrow left parenthesis 3 comma 2 right parenthesis space space which space is space not space belongs space to space straight R.
The space number space of space relations space containing space left parenthesis 1 comma 2 right parenthesis space and space left parenthesis 1 comma 3 right parenthesis space
which space are space reflexive space and space symmentric space but space not space transitive space is space 1. end style

Question 9

The relation 'R' in N × N such that (a, b) R (c, d) begin mathsize 12px style left right double arrow end style 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

begin mathsize 12px style Correct space option colon space left parenthesis straight c right parenthesis
Given space straight R space is space left parenthesis straight a comma straight a right parenthesis space straight R space left parenthesis straight c comma space straight c right parenthesis space rightwards double arrow straight a space plus space straight d space equals space straight b space plus space straight c space
For space space left parenthesis straight a comma space straight a right parenthesis space straight R space open parentheses straight b comma space straight b close parentheses rightwards double arrow straight a space plus space straight b space equals straight a space plus space straight b
For space space left parenthesis straight b comma space straight a right parenthesis space straight R space left parenthesis straight d comma space straight c right parenthesis rightwards double arrow space straight b space plus space straight c space equals straight a plus straight d
For space open parentheses straight a comma straight b close parentheses comma space open parentheses straight c comma straight d close parentheses space and space open parentheses straight u comma straight v close parentheses
straight a plus straight d equals straight b plus straight c space and space straight c plus straight v equals straight d plus straight u
adding space RHS space and space LHS space of space the space above space equations comma
straight a plus straight d plus straight c plus straight v equals straight b plus straight c plus straight d plus straight u
straight a plus straight v equals straight b plus straight u
rightwards double arrow open parentheses straight a comma straight b close parentheses space straight R space open parentheses straight u comma straight v close parentheses
Here comma space straight R space follows space reflexive comma space symmetric space and space transitive space properties.
hence comma space it space is space equivalence space relation. end style

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

begin mathsize 12px style Correct space option colon space left parenthesis straight c right parenthesis
Given space that space straight A space equals space left curly bracket 1 comma space 2 comma space 3 right curly bracket comma space straight B space equals space left curly bracket 1 comma space 4 comma space 6 comma space 9 right curly bracket space and
straight R space is space straight a space relation space from space straight A space to space straight B space defined space by space apostrophe straight x space is space greater space than space straight y apostrophe
Hence space straight R space equals open curly brackets open parentheses 2 comma 1 close parentheses comma open parentheses 3 comma 1 close parentheses close curly brackets
Range space of space straight y space is space open curly brackets 1 close curly brackets.

end style

Question 11

A relation R is defined from {2, 3, 4, 5} to {3, 6, 7, 10} by : x R y begin mathsize 12px style left right double arrow end style 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

begin mathsize 12px style Correct space option colon space left parenthesis straight d right parenthesis
Given space that space relation space straight R space is space defined space from space left curly bracket 2 comma space 3 comma space 4 comma space 5 right curly bracket space to space left curly bracket 3 comma space 6 comma space 7 comma space 10 right curly bracket space by space colon space straight x space straight R space straight y space straight x space is space relatively space prime space to space straight y.
straight R space can space be space written space as
open curly brackets open parentheses 2 comma 3 close parentheses comma open parentheses 2 comma 7 close parentheses comma open parentheses 3 comma 7 close parentheses comma open parentheses 3 comma 10 close parentheses comma open parentheses 4 comma 3 close parentheses comma open parentheses 4 comma 7 close parentheses comma open parentheses 5 comma 3 close parentheses comma open parentheses 5 comma 6 close parentheses comma open parentheses 5 comma 7 close parentheses close curly brackets
Here space we space can space see space that space domain space means space straight x space element space which space is space 2 less or equal than straight x less or equal than 5.
Hence comma space open curly brackets 2 comma 3 comma 4 comma 5 close curly brackets
end style

Question 12

A relation Φ from C to R is defined by  begin mathsize 12px style straight x space straight ϕ space straight y space left right double arrow open vertical bar straight x close vertical bar equals straight y. end style Which one is correct ?

 begin mathsize 12px style open parentheses straight a close parentheses space open parentheses 2 plus 3 straight i close parentheses straight ϕ 13
open parentheses straight b close parentheses space 3 straight ϕ open parentheses negative space 3 close parentheses
open parentheses straight c close parentheses space open parentheses 1 space plus space straight i close parentheses space straight ϕ space 2
open parentheses straight d close parentheses space straight i space straight ϕ space 1 end style

Solution 12

begin mathsize 12px style Correct space option colon left parenthesis straight d right parenthesis
Given space that space relation space straight capital phi space from space straight C space to space straight R space is space defined space by space straight x space straight ϕ space straight y space left right double arrow open vertical bar straight x close vertical bar equals straight y
For space straight i space straight ϕ space 1 comma
open vertical bar straight i close vertical bar equals straight i squared equals negative 1 space where space straight i space is space straight a space complex space number.
Also comma space open vertical bar negative 1 close vertical bar equals 1
Hence comma space straight i space straight ϕ space 1 space is space correct. end style

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

begin mathsize 12px style Correct space option colon space left parenthesis straight c right parenthesis
Given space that space straight R space be space straight a space relation space on space straight N space defined space by space straight x space plus space 2 straight y space equals space 8.
Here comma space straight x equals 8 space will space give space us space straight y equals 0. space It space does space not space satisfy space the space condition space on space straight N.
Hence comma space options space left parenthesis straight a right parenthesis comma space left parenthesis straight b right parenthesis space will space exclude.
For space option space left parenthesis straight d right parenthesis space straight x equals 1 space gives space you space straight y equals 7 over 2 space which space is space not space natural space number.
The space answer space is space open curly brackets 2 comma 4 comma 6 close curly brackets. end style

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

begin mathsize 12px style
Corr ect space option colon left parenthesis straight a right parenthesis
Given space that space straight R space is space straight a space relation space from space left curly bracket 11 comma space 12 comma space 13 right curly bracket space to space left curly bracket 8 comma space 10 comma space 12 right curly bracket space defined space by space straight y space equals space straight x space minus space 3.
straight R equals open curly brackets open parentheses 11 comma 8 close parentheses comma open parentheses 13 comma 10 close parentheses close curly brackets
straight R to the power of negative 1 end exponent equals open curly brackets open parentheses 8 comma 11 close parentheses comma open parentheses 10 comma 13 close parentheses close curly brackets

As space inverse space function space of space straight R space is space
straight y plus 3 equals straight x
rightwards double arrow straight y equals straight x plus 3 end style

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

begin mathsize 12px style Correct space option colon left parenthesis straight b right parenthesis
Given space that space straight R space equals space left curly bracket left parenthesis straight a comma space straight a right parenthesis comma space left parenthesis straight b comma space straight b right parenthesis comma space left parenthesis straight c comma space straight c right parenthesis comma space left parenthesis straight a comma space straight b right parenthesis right curly bracket space be space straight a space relation space on space set space straight A space equals space left curly bracket straight a comma space straight b comma space straight c right curly bracket.
Here comma space left parenthesis straight a comma space straight a right parenthesis comma space left parenthesis straight b comma space straight b right parenthesis comma space left parenthesis straight c comma space straight c right parenthesis space represents space straight a space reflexive space relation.
As space space left parenthesis straight a comma space straight b right parenthesis space element of straight R space but space open parentheses straight b comma straight a close parentheses space not space belongs space to space straight R.
Hence comma space it space is space not space symmetric. end style

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

begin mathsize 12px style Correct space option colon left parenthesis straight c right parenthesis
Given space that space space straight A space equals space left curly bracket 1 comma space 2 comma space 3 right curly bracket space and space straight R space equals space left curly bracket left parenthesis 1 comma space 2 right parenthesis comma space left parenthesis 2 comma space 3 right parenthesis comma space left parenthesis 1 comma space 3 right parenthesis right curly bracket space be space straight a space relation space on space straight A. space
We space can space see space clearly space that space it space is space neither space reflexive space nor space symmetric space but space transitive.
end style

Question 17

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

begin mathsize 12px style left parenthesis straight a right parenthesis space straight R space subset of space straight S end style

begin mathsize 12px style left parenthesis straight b right parenthesis space straight S space subset of space straight R end style

begin mathsize 12px style left parenthesis straight c right parenthesis space straight R space equals space straight S end style

begin mathsize 12px style left parenthesis straight d right parenthesis space none space of space these end style

Solution 17

begin mathsize 12px style Correct space option colon space left parenthesis straight b right parenthesis
Given space that space straight R space is space the space largest space relation space on space straight A space and space straight S space is space any space relation space on space straight A.
We space know space that space straight R space is space always space subset space of space straight A cross times straight A.
hence comma space straight S subset of straight R. end style

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 begin mathsize 12px style left right double arrow end style 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

begin mathsize 12px style Correct space option colon left parenthesis straight d right parenthesis
Given space that space straight R space is space straight a space relation space on space the space set space straight A space equals space left curly bracket 1 comma space 2 comma space 3 comma space 4 comma space 5 comma space 6 comma space 7 comma space 8 comma space 9 right curly bracket space
given space by space straight x space straight R space straight y space rightwards double arrow straight y space equals space 3 straight x.
This space means space second space element space should space be space thrice space of space the space first.
It space is space clear space that space option space left parenthesis straight a right parenthesis comma space left parenthesis straight b right parenthesis comma space left parenthesis straight c right parenthesis space does space not space satisfy space the space given space condition.
Hence comma space option space is space left parenthesis straight d right parenthesis.
end style

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

begin mathsize 12px style Correct space option colon space left parenthesis straight d right parenthesis
Given space that space straight R space is space straight a space Relation space on space the space set space straight A space equals space left curly bracket 1 comma space 2 comma space 3 right curly bracket space given space by space straight R space equals space left curly bracket left parenthesis 1 comma space 1 right parenthesis comma space left parenthesis 2 comma space 2 right parenthesis comma space left parenthesis 3 comma space 3 right parenthesis right curly bracket
By space definition space straight R space is space reflexive space and space symmetric.
If space we space take space left parenthesis 1 comma space 1 right parenthesis space with space space then space we space can space see space that space transitivity space followed.
Answer space is space all space the space three space options.
end style

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

begin mathsize 12px style Correct space option colon left parenthesis straight a right parenthesis
Given space that space space straight A space equals space left curly bracket straight a comma space straight b comma space straight c comma space straight d right curly bracket space then space straight a space relatio n space straight R space equals space left curly bracket left parenthesis straight a comma space straight b right parenthesis comma space left parenthesis straight b comma space straight a right parenthesis comma space left parenthesis straight a comma space straight a right parenthesis right curly bracket space on space straight A.
left parenthesis straight a comma space straight b right parenthesis comma space left parenthesis straight b comma space straight a right parenthesis space element of straight R space rightwards double arrow straight R space is space symmetric.
Also space for space open parentheses straight a comma straight a close parentheses space straight R space is space symmetric.

end style

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

begin mathsize 12px style Correct space option colon space left parenthesis straight c right parenthesis
As space only space one space element space is space in space straight R comma space it space is space not space reflexive space or space symmetric.
Hence comma space it space is space trannsitive space only. end style

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

begin mathsize 12px style Correct space option colon space left parenthesis straight b right parenthesis
Given space that space straight R equals space left curly bracket left parenthesis 1 comma space 2 right parenthesis comma space left parenthesis 2 comma space 2 right parenthesis comma space left parenthesis 1 comma space 1 right parenthesis comma space left parenthesis 4 comma space 4 right parenthesis comma space left parenthesis 1 comma space 3 right parenthesis comma space left parenthesis 3 comma space 3 right parenthesis comma space left parenthesis 3 comma space 2 right parenthesis right curly bracket
straight R space is space reflexive space as space straight R space contains space open parentheses 2 comma 2 close parentheses space and space open parentheses 3 comma 3 close parentheses.
straight R space is space transitive space as space straight R space contains space open parentheses 1 comma space 2 close parentheses space and space left parenthesis 2 comma space 2 right parenthesis space element of straight R space rightwards double arrow open parentheses 1 comma space 2 close parentheses element of straight R.
Hence comma space straight R space is space reflexive space and space transitive space but space not space symmetric. end style

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

begin mathsize 12px style Correct space option colon space left parenthesis straight b right parenthesis
Given space that space straight A space equals space left curly bracket 1 comma space 2 comma space 3 right curly bracket. space Then comma space the space number space of space equivalence space relation space containing space left parenthesis 1 comma space 2 right parenthesis space is
straight R subscript 1 equals open curly brackets open parentheses 1 comma 1 close parentheses comma open parentheses 2 comma 2 close parentheses comma open parentheses 3 comma 3 close parentheses comma open parentheses 1 comma 2 close parentheses comma open parentheses 2 comma 1 close parentheses close curly brackets
straight R subscript 2 equals open curly brackets open parentheses 1 comma 1 close parentheses comma open parentheses 2 comma 2 close parentheses comma open parentheses 3 comma 3 close parentheses comma open parentheses 1 comma 2 close parentheses comma open parentheses 2 comma 1 close parentheses comma open parentheses 2 comma 3 close parentheses comma open parentheses 3 comma 2 close parentheses comma open parentheses 1 comma 3 close parentheses comma open parentheses 3 comma 1 close parentheses close curly brackets
Then comma space the space number space of space equivalence space relation space containing space left parenthesis 1 comma space 2 right parenthesis space is space 2.
end style

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

begin mathsize 12px style Correct space option colon left parenthesis straight c right parenthesis
Given space straight R equals left curly bracket left parenthesis 1 comma space 1 right parenthesis comma space left parenthesis 2 comma space 2 right parenthesis comma space left parenthesis 3 comma space 3 right parenthesis right curly bracket space on space the space set space left curly bracket 1 comma space 2 comma space 3 right curly bracket
It space is space clear space that space it space is space reflexive space and space symmetric space by space definition.
It space is space also space transitive.
end style

Question 25

S is relation over the set R of all real numbers and it is given by (a, b) ε S begin mathsize 12px style left right double arrow end style ab ≥ 0. Then, S is

(a) symmetric and transitive only

(b) reflexive and symmetric only

(c) antisymmetric relation

(d) an equivalence relation

Solution 25

begin mathsize 12px style Correct space option colon space left parenthesis straight b right parenthesis
Given space relation space over space the space set space straight R space of space all space real space numbers space and space it space is space given space by space open parentheses straight a comma straight b close parentheses element of straight S space left right double arrow space ab greater or equal than 0
Hence comma space ab greater or equal than 0 space if space and space only space if space either space both space are space positive space or space negative.
As space left parenthesis straight a comma straight a right parenthesis element of straight S space rightwards double arrow straight a squared greater or equal than 0 space it space is space reflexive.
As space left parenthesis straight a comma straight b right parenthesis element of straight S space rightwards double arrow ab greater or equal than 0 space also space ba greater or equal than 0 space it space is space symmetric.
end style

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

begin mathsize 12px style Correct space option colon space left parenthesis straight a right parenthesis
In space the space set space of space straight Z space of space all space integers space straight x space straight R space straight y space colon space if space straight x less or equal than straight y space is space not space an space equivalence space relation.
For space the space relation space space straight x less or equal than straight y space open parentheses straight x comma space straight y close parentheses element of straight R space but space open parentheses straight y comma space straight x close parentheses space not space belongs space to space straight y space as space straight y greater or equal than straight x space given.
Hence comma space it space is space not space an space equivalence space relation.
end style

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

begin mathsize 12px style Correct space option colon left parenthesis straight a right parenthesis
Given space that space straight R equals left curly bracket left parenthesis 1 comma space 1 right parenthesis comma space left parenthesis 2 comma space 2 right parenthesis comma space left parenthesis 3 comma space 3 right parenthesis comma space left parenthesis 1 comma space 2 right parenthesis comma space left parenthesis 2 comma space 3 right parenthesis comma space left parenthesis 1 comma space 3 right parenthesis right curly bracket
From space space left parenthesis 1 comma space 1 right parenthesis comma space left parenthesis 2 comma space 2 right parenthesis comma space left parenthesis 3 comma space 3 right parenthesis space it space is space reflexive.
For space left parenthesis 1 comma 2 right parenthesis comma space left parenthesis 2 comma space 3 right parenthesis comma space left parenthesis 1 comma space 3 right parenthesis space space do space not space have space open parentheses 2 comma 1 close parentheses comma space open parentheses 3 comma 2 close parentheses comma space open parentheses 3 comma 1 close parentheses space respectively.
end style

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

begin mathsize 12px style Correct space option colon space left parenthesis straight d right parenthesis
The space maximum space number space of space equivalence space relations space on space the space set space straight A space equals space left curly bracket 1 comma space 2 comma space 3 right curly bracket space is
straight R subscript 1 equals open curly brackets open parentheses 1 comma 1 close parentheses close curly brackets
straight R subscript 2 equals open curly brackets open parentheses 2 comma 2 close parentheses close curly brackets
straight R subscript 3 equals open curly brackets open parentheses 3 comma 3 close parentheses close curly brackets
straight R subscript 4 equals open curly brackets open parentheses 1 comma 1 close parentheses comma open parentheses 2 comma 2 close parentheses comma open parentheses 3 comma 3 close parentheses comma open parentheses 1 comma 2 close parentheses comma open parentheses 2 comma 1 close parentheses close curly brackets
straight R subscript 5 equals open curly brackets open parentheses 1 comma 1 close parentheses comma open parentheses 2 comma 2 close parentheses comma open parentheses 3 comma 3 close parentheses comma open parentheses 1 comma 2 close parentheses comma open parentheses 2 comma 1 close parentheses comma open parentheses 1 comma 3 close parentheses comma open parentheses 3 comma 1 close parentheses comma open parentheses 2 comma 3 close parentheses comma open parentheses 3 comma 2 close parentheses close curly brackets
The space maximum space number space of space equivalence space relations space on space the space set space straight A space equals space left curly bracket 1 comma space 2 comma space 3 right curly bracket space is space 5.
end style

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

begin mathsize 12px style Correct space option colon space left parenthesis straight d right parenthesis
straight R space be space straight a space relation space on space the space set space straight N space of space straight a space natural space numbers space defined space by space straight n space straight R space straight m space iff space straight n space divides space straight m.
straight R space is space reflexive space as space straight m space divides space straight m space but space it space is space not space symmetric space as space we space can space not space say space that
straight n space divides space straight m space rightwards double arrow straight m space divides space straight n.
straight R space is space transitive space as space straight a space divides space straight b space and space straight b space divides space straight c space rightwards double arrow straight a space divides space straight c. end style

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

begin mathsize 12px style Correct space option colon space left parenthesis straight b right parenthesis
Given space that space straight L space denote space the space set space of space all space straight space lines space in space straight a space plane. space
straight A space relation space straight R space be space defined space by space straight l space straight R space straight m space iff space straight l space is space perpendicular space to space straight m space for space all space straight l comma space straight m space straight epsilon space straight L.
straight R space is space not space reflexive. space straight R space is space symmetric space as space we space can space say space l space perpendicular straight m space or space straight m perpendicular l italic. end style

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

begin mathsize 12px style Correct space option colon space left parenthesis straight c right parenthesis
Given space that space straight R space is space straight T space be space the space set space of space all space triangles space in space the space Euclidean space plane comma space
and space straight a space relation space straight R space on space straight T space be space defined space as space straight a space straight R space straight b space if space straight a space is space congruent space to space straight b space for space all space straight a comma space straight b space straight epsilon space straight T.
Here comma space congruency space of space triangles space follows space reflexive comma space symmetric space and space transitive space property.
Hence comma space it space is space an space equivalence space relation. end style

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

begin mathsize 12px style Correct space option colon space left parenthesis straight b right parenthesis
Given space that space straight a space non minus empty space set space consisting space of space children space in space straight a space family space and space straight a space relation space
straight R space defined space as space straight a space straight R space straight b space if space straight a space is space brother space of space straight b.
Here comma space straight a space straight R space straight b space is space not space symmetric space in space case space of space brother space and space sister space straight a space straight R space straight b space satisfies space but space straight b space straight R space straight a space does space not space satisfy.
Hence comma space it space is space transitive space but space not space symmetric. end style

Question 33

For real numbers x and y, define x R y if x - y + begin mathsize 12px style square root of 2 end style is an irrational number. Then the relation R is

(a) reflexive

(b) symmetric

(c) transitive

(d) none of these

Solution 33

begin mathsize 12px style Correct space option colon space left parenthesis straight a right parenthesis
Given space that space For space real space numbers space straight x space and space straight y comma space define space straight x space straight R space straight y space if space straight x space minus space straight y space plus square root of 2 is space an space irrational space number. space
straight R space is space reflexive space as space for space straight x equals straight y space semicolon space space straight x space minus space straight y space plus square root of 2 is space an space irrational space number. space
end style

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

G i v e n space t h a t space open parentheses 1 comma 2 close parentheses element of R space a n d space open parentheses 2 comma 1 close parentheses element of R.
S o space f o r space a space r e l a t i o n space R space t o space b e space t r a n s i t i v e comma space open parentheses 1 comma 2 close parentheses element of R space a n d space open parentheses 2 comma 1 close parentheses element of R rightwards double arrow open parentheses 1 comma 1 close parentheses element of R.
B u t space open parentheses 1 comma 1 close parentheses space i s space n o t space a n space e l e m e n t space o f space R.

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}