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

## Chapter 1 - Relations Exercise MCQ

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

a. Reflexive

b. Transitive

c. Symmetric

d. None of these

Given: R = {(1, 2)}

R is not reflexive as (1, 1), (2, 2), (3, 3) ∉ R.

Since (1, 2) ∈ R but (2, 1) ∉ R

Therefore, R is not symmetric.

Also, R is not transitive.

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

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

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

(a) reflexive and transitive

(b) reflexive and symmetric

(c) symmetric and transitive

(d) an equivalence relation

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

Let R be the relation over the set of all straight lines in a plane such that l_{1} R l_{2} l_{1}l2. Then, R is

(a) symmetric

(b) reflexive

(c) transitive

(d) an equivalence relation

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

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

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

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

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

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}

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

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}

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

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

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

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

_{}

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

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

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

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

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

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

(a) 1

(b) 2

(c) 3

(d) 4

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

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

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)

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

The relation S defined on the set R of all real number by the rule a Sb iff ab≥ 0 is

(a) symmetric and transitive only

(b) reflexive and symmetric only

(c) antisymetric relation

(d) an equivalence relation

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

(a) 1

(b) 2

(c) 3

(d) 5

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

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

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

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

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

## Chapter 1 - Relations Exercise Ex. 1.1

Three relation R_{1}, R_{2}, R_{3} and R_{4} defined on a set A = {a,b,c} as follows:

R_{1} = {(a, a), (a, b), (a, c), (b, b), (b, c), (c, a), (c, b), (c, c)}.

R_{2} = {(a, a)}

R_{3} = {(b, c)}

R_{4} = {(a, b), (b, c), (c, a)}

Find whether or not each of the relations R_{1}, R_{2}, R_{3}, R_{4} on A is (i) reflexive (ii) symmetric (iii) transitive.

Check wthether the relation R on **R** defined as R = {(a, b): a ≤ b^{3}} is reflexive, symmetric or transitive.

Give an Example of a relation which is

- Reflexive and symmetric but not transitive
- Reflexive and transitive but not symmetric
- Symmetric and transitive but not reflexive
- Symmetric but neither reflexive nor transitive
- Transitive but neither reflexive nor symmetric

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 ]

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.

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 ]

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 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.

Check if the relation R in the set of all real numbers defined as R = {(a, b) : a < b} is

(i) symmetric (ii) transitive

Given: R = {(a, b) : a < b}

Clearly, 1 < 2

That is, (1, 2) ∈ R

But 2 ≮ 1

So, (2, 1) ∉ R

Therefore, R is not symmetric.

Let (a, b), (b, c) ∈ R

∴ a < b and b < c

∴ a < c

Therefore, (a, c) ∈ R

Thus, R is transitive.

Let W denote the set of words in English dictionary. Define the relation R by R = {(x, y) ∈ W × W such that x and y have at least one letter in common}. Show that this relation R is reflexive and symmetric but not transitive.

Given: R = {(x, y) ∈ W × W such that x and y have at least one letter in common}

Clearly, (x, x) ∈ R as x and x have all the letters common.

Therefore, R is reflexive.

Now suppose, x and y have at least one letter in common.

So, y and x will also have at least one letter in common.

Therefore, R is symmetric.

Now take, x = Sun, y = Moon and z = Mars

Here, (x, y) ∈ R and (y, z) ∈ R as x and y have the letter "n" common whereas y and z have the letter "m" common.

But, x and z do not have any letter common.

Therefore, (x, z) ∉ R.

Thus, R is not transitive.

## Chapter 1 - Relations Exercise Ex. 1.2

Check whether the relation R on the set N of natural numbers given by R = {(a, b): a is divisor of b} is reflexive, symmetric or transitive. Also, determine whether R is an equivalence relation.

Given: R = {(a, b): a is divisor of b}

Any natural number "a" divides itself.

So, a is divisor of a.

Therefore, R is reflexive.

We know that 2 is divisor of 4.

So, (2, 4) ∈ R

But 4 does not divide 2, then (4, 2) ∉ R

Therefore, R is not symmetric.

As R is not symmetric.

Thus, R is not an equivalence relation.

Let
C be the set of all complex numbers and C_{0} be the set of all
non-zero complex numbers. Let a relation R on C_{0} be defined as

z_{1}
R z_{2}⇔ is real for
all z_{1}, z_{2}∈ C_{0}. Show that R is an
equivalence relation.

To check reflexive:

Now, which is real.

So, z_{1}
R z_{1}.

Therefore, R is reflexive.

To check symmetric:

Suppose z_{1}
R z_{2}

is real

Let where k is real.

Now,

Therefore, is real as -k is real.

Therefore, R is symmetric.

To check transitive:

Suppose z_{1}
R z_{2} and z_{2} R z_{3}

are real

Let

… (i)

Now, which is real.

Thus, R is transitive.

Hence, R is an equivalence relation.

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.

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

R = {(p_{1}, p_{2}) : p_{1} and p_{2} 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?

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}.

## Chapter 1 - Relations Exercise Ex. 1VSAQ

State the reason for the relation R on the set {1, 2, 3} given by R = {(1, 2), (2, 1)} not to be transitive.

Let
*R* = {(*a*, *a*^{3}): *a* is a prime
number less than 5} be a relation. Find the range of *R*.

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

∴ The range set of R is {8, 27}.

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 ]

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,...}

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.

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 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.

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?

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 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.

Let the relation R be defined on the set

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

Write R as a set of ordered pairs.

A = {1, 2, 3, 4, 5} and R = {(a, b) :|a^{2}
- b^{2} | < 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)}

Let the relation R be define on N by aRb iff

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

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)}

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

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 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.

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