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#### Question 1:

Let A be the set of all human beings in a town at a particular time. Determine whether each of the following relations are reflexive, symmetric and transitive:

(i) R = {(x, y) : x and y work at the same place}
(ii) R = {(x, y) : x and y live in the same locality}
(iii) R = {(x, y) : x is wife of y}
(iv) R = {(x, y) : x is father of and y}

(i) Reflexivity:

Symmetry:

Transitivity:

(ii) Reflexivity:

Symmetry:

Transitivity:

(iii)
Reflexivity:

Symmetry:

Transitivity:

(iv)
Reflexivity:

Symmetry:

Transitivity:

#### Question 2:

Three relations R1, R2 and R3 are defined on a set A = {a, b, c} as follows:
R1 = {(a, a), (a, b), (a, c), (b, b), (b, c), (c, a), (c, b), (c, c)}
R2 = {(a, a)}
R3 = {(b, c)}
R4 = {(a, b), (b, c), (c, a)}.

Find whether or not each of the relations R1, R2, R3, R4 on A is (i) reflexive (ii) symmetric and (iii) transitive.

(i) R1
Reflexive:
Clearly, (a, a), (b, b) and (c, c)$\in$R1
So, R1 is reflexive.

Symmetric:
We see that the ordered pairs obtained by interchanging the components of R1 are also in R1.
So, R1 is symmetric.

Transitive:
Here,

So, R1 is transitive.

(ii) R2
Reflexive: Clearly $\left(a,a\right)\in {R}_{2}$. So, R2 is reflexive.
Symmetric: Clearly $\left(a,a\right)\in R⇒\left(a,a\right)\in R$. So, R2 is symmetric.
Transitive: R2 is clearly a transitive relation, since there is only one element in it.

(iii) R3
Reflexive:
Here,

So, R3  is not reflexive.

Symmetric:
Here,

Transitive:
Here, R3 has only two elements. Hence, R3 is transitive.

(iv) R4
Reflexive:
Here,

Symmetric:
Here,

Transitive:
Here,

#### Question 3:

Test whether the following relations R1, R2, and R3 are (i) reflexive (ii) symmetric and (iii) transitive:
(i) R1 on Q0 defined by (a, b) ∈ R1a = 1/b.
(ii) R2 on Z defined by (a, b) ∈ R2 ⇔ |a – b| ≤ 5
(iii) R3 on R defined by (a, b) ∈ R3a2 – 4ab + 3b2 = 0.

(i) Reflexivity:
Let a be an arbitrary element of R1. Then,

Symmetry:
Let (a, b) $\in {R}_{1}$. Then,

Transitivity:
Here,

(ii)
Reflexivity:
Let a be an arbitrary element of R2. Then,

Symmetry:

Transitivity:

(iii)
Reflexivity: Let a be an arbitrary element of R3. Then,

Symmetry:

Transitivity:

#### Question 4:

Let A = {1, 2, 3}, and let R1 = {(1, 1), (1, 3), (3, 1), (2, 2), (2, 1), (3, 3)}, R2 = {(2, 2), (3, 1), (1, 3)}, R3 = {(1, 3), (3, 3)}. Find whether or not each of the relations R1, R2, R3 on A is (i) reflexive (ii) symmetric (iii) transitive.

$\left(1\right)$ R1
Reflexivity:
Here,

Symmetry:

Transitivity:

$\left(2\right)$ R2
Reflexivity:

Symmetry:

Transitivity:

$\left(3\right)$ R3
Reflexivity:

Symmetry:

Transitivity:

#### Question 5:

The following relations are defined on the set of real numbers.
(i) aRb if ab > 0
(ii) aRb if 1 + ab > 0
(iii) aRb if |a| ≤ b

Find whether these relations are reflexive, symmetric or transitive.

(i)
Reflexivity: Let a be an arbitrary element of R. Then,

Symmetry:

Transitivity:

(ii)
Reflexivity: Let a be an arbitrary element of R. Then,

Symmetry:

Transitivity:

(iii)
Reflexivity: Let a be an arbitrary element of R. Then,

Symmetry:

Transitivity:

#### Question 6:

Check whether the relation R defined on the set A = {1, 2, 3, 4, 5, 6} as R = {(a, b) : b = a + 1} is reflexive, symmetric or transitive.

Reflexivity:

Symmetry:

Transitivity:

#### Question 7:

Check whether the relation R on R defined by R = {(a, b) : ab3} is reflexive, symmetric or transitive.

Reflexivity:

Symmetry:

Transitivity:

#### Question 8:

Prove that every identity relation on a set is reflexive, but the converse is not necessarily true.

Let A be a set. Then,

The converse of it need not be necessarily true.
Consider the set A = {1, 2, 3}

Here,
Relation R = {(1, 1), (2, 2) , (3, 3), (2, 1), (1, 3)} is reflexive on A.
However, R is not an identity relation.

#### Question 9:

If A = {1, 2, 3, 4} define relations on A which have properties of being
(i) reflexive, transitive but not symmetric
(ii) symmetric but neither reflexive nor transitive
(iii) reflexive, symmetric and transitive.

(i) The relation on A having properties of being reflexive, transitive, but not symmetric is
R = {(1, 1), (2, 2), (3, 3), (4, 4), (2, 1)}

(ii) The relation on A having properties of being symmetric, but neither reflexive nor transitive is
R = {(1, 2), (2, 1)}
The relation R on A is neither reflexive nor transitive, but symmetric.

(iii) The relation on A having properties of being symmetric, reflexive and transitive is
R = {(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 1)}
The relation R is an equivalence relation on A.

#### Question 10:

Let R be a relation defined on the set of natural numbers N as
R = {(x, y) : x, y N, 2x + y = 41}
Find the domain and range of R. Also, verify whether R is (i) reflexive, (ii) symmetric (iii) transitive.

Domain of R is the values of x and range of R is the values of y that together should satisfy 2x+y = 41.
So,
Domain of R = {1, 2, 3, 4, ... , 20}
Range of R = {1, 3, 5, ... , 37, 39}

Reflexivity: Let x be an arbitrary element of R. Then,

Symmetry:

Transitivity:

#### Question 11:

Is it true that every relation which is symmetric and transitive is also reflexive? Give reasons.

No, it is not true.

Consider a set A = {1, 2, 3} and relation R on A such that R = {(1, 2), (2, 1), (2, 3), (1, 3)}
The relation R on A is symmetric and transitive. However, it is not reflexive.

Hence, R is not reflexive.

#### Question 12:

An integer m is said to be related to another integer n if m is a multiple of n.Check if the relation is symmetric, reflexive and transitive.

#### Question 13:

Show that the relation '≥' on the set R of all real numbers is reflexive and transitive but not symmetric.

Let R be the set such that R = {(a, b) : a, b}

Reflexivity:

Symmetry:

Transitivity:

#### Question 14:

Give an example of a relation which is
(i) reflexive and symmetric but not transitive;
(ii) reflexive and transitive but not symmetric;
(iii) symmetric and transitive but not reflexive;
(iv) symmetric but neither reflexive nor transitive.
(v) transitive but neither reflexive nor symmetric.

Suppose A be the set such that A = {1, 2, 3}

(i) Let R be the relation on A such that
R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3)}
Thus,
R is reflexive and symmetric, but not transitive.

(ii) Let R be the relation on A such that
R = {(1, 1), (2, 2), (3, 3), (1, 2), (1, 3), (2, 3)}
Clearly, the relation R on A is reflexive and transitive, but not symmetric.

(iii) Let R be the relation on A such that
R = {(1, 2), (2, 1), (1, 3), (3, 1), (2, 3)}
We see that the relation R on A is symmetric and transitive, but not reflexive.

(iv) Let R be the relation on A such that
R = {(1, 2), (2, 1), (1, 3), (3, 1)}
The relation R on A is symmetric, but neither reflexive nor transitive.

(v) Let R be the relation on A such that
R = {(1, 2), (2, 3), (1, 3)}
The relation R on A is transitive, but neither symmetric nor reflexive.

#### Question 15:

Given the relation R = {(1, 2), (2, 3)} on the set A = {1, 2, 3}, add a minimum number of ordered pairs so that the enlarged relation is symmeteric, transitive and reflexive.

We have,

R = {(1, 2), (2, 3)}

R can be a transitive only when the elements (1, 3) is added

R can be a reflexive only when the elements (1, 1), (2, 2), (3, 3) are added

R can be a symmetric only when the elements (2, 1), (3, 1) and (3, 2) are added

So, the required enlarged relation, R' = {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)} = A $×$ A

#### Question 16:

Let A = {1, 2, 3} and R = {(1, 2), (1, 1), (2, 3)} be a relation on A. What minimum number of ordered pairs may be added to R so that it may become a transitive relation on A.

We have,

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

To make R a transitive relation on A, (1, 3) must be added to it.

So, the minimum number of ordered pairs that may be added to R to make it a transitive relation is 1.

#### 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]

We have,

A = {abc} and R = {(aa), (bc), (ab)}

R can be a reflexive relation only when elements (b, b) and (c, c) are added to it

R can be a transitive relation only when the element (ac) is added to it

So, the minmum number of ordered pairs to be added in R is 3.

#### Question 18:

Each of the following defines a relation on N:

(i) x > y, x, y $\in$ N
(ii) x + y = 10, xy $\in$ N
(iii) xy is square of an integer, x, y $\in$ N
(iv) x + 4y = 10, x, y $\in$ N

Determine which of the above relations are reflexive, symmetric and transitive.                                                            [NCERT EXEMPLAR]

(i) We have,

R = {(x, y) : x > yxy $\in$ N}

(ii) We have,

R = {(x, y) : x + y = 10, xy $\in$ N}

(iii) We have,

R = {(x, y) : xy is square of an integer, xy $\in$ N}

(iv) We have,

R = {(x, y) : x + 4y = 10, xy $\in$ N}

#### Question 1:

Show that the relation R defined by R = {(a, b) : a – b is divisible by 3; a, b Z} is an equivalence relation.

We observe the following relations of relation R.

Reflexivity:

Symmetry:

Transitivity:

Hence, R is an equivalence relation on Z.

#### Question 2:

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

We observe the following properties of relation R.

Reflexivity:

Symmetry:

Transitivity:

Hence, R is an equivalence relation on Z.

#### Question 3:

Prove that the relation R on Z defined by
(a, b) ∈ Rab is divisible by 5
is an equivalence relation on Z.

We observe the following properties of relation R.

Reflexivity:

Symmetry:

Transitivity:

Hence, R is an equivalence relation on Z.

#### Question 4:

Let n be a fixed positive integer. Define a relation R on Z as follows:
(a, b) ∈ Rab is divisible by n.
Show that R is an equivalence relation on Z.

We observe the following properties of R. Then,
Reflexivity:

Symmetry:

Transitivity:

Hence, R is an equivalence relation on Z.

#### Question 5:

Let Z be the set of integers. Show that the relation
R = {(a, b) : a, bZ and a + b is even}
is an equivalence relation on Z.

We observe the following properties of R.

Reflexivity:

Symmetry:

Transitivity:

Hence, R is an equivalence relation on Z.

#### Question 6:

m is said to be related to n if m and n are integers and mn is divisible by 13. Does this define an equivalence relation?

We observe the following properties of relation R.

Hence, R is an equivalence relation on Z.

#### Question 7:

Let R be a relation on the set A of ordered pair of integers defined by (x, y) R (u, v) if xv = yu. Show that R is an equivalence relation.

We observe the following properties of R.

Hence, R is an equivalence relation on A.

#### Question 8:

Show that the relation R on the set A = {x Z ; 0 ≤ x ≤ 12}, given by R = {(a, b) : a = b}, is an equivalence relation. Find the set of all elements related to 1.

We observe the following properties of R.

Reflexivity: Let a be an arbitrary element of A. Then,

Hence, R is an equivalence relation on A.

The set of all elements related to 1 is {1}.

#### Question 9:

Let L be the set of all lines in XY-plane and R be the relation in L defined as R = {L1, L2) : L1 is parallel to L2}. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4.

We observe the following properties of R.

Hence, R is an equivalence relation on L.

Set of all the lines related to y = 2x+4
= L' = {(x, y) : y = 2x+c, where c$\in$R}

#### 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?

We observe the following properties on R.

Hence, R is an equivalence relation on the set A.

Also, the set of all the triangles$\in$A is related to the right angle triangle T with the sides 3, 4, 5.

#### Question 11:

Let O be the origin. We define a relation between two points P and Q in a plane if OP = OQ. Show that the relation, so defined is an equivalence relation.

Let A be the set of all points in a plane such that

We observe the following properties of R.

Reflexivity: Let P be an arbitrary element of R.
The distance of a point P will remain the same from the origin.
So, OP = OP

Hence, R is an equivalence relation on A.

#### 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}.

We observe the following properties of R.

Reflexivity:

Thus, R is an equivalence relation on A.

We observe that all the elements of the subset {1, 3, 5, 7} are odd. Thus, they are related to each other.
This is because the relation R on A is an equivalence relation.

Similarly, the elements of the subset {2, 4, 6} are even. Thus, they are related to each other because every element is even.

Hence proved.

#### Question 13:

Let S be a relation on the set R of all real numbers defined by
S = {(a, b) ∈ R × R : a2 + b2 = 1}
Prove that S is not an equivalence relation on R.

We observe the following properties of S.

Hence, S is not an equivalence relation on R.

#### Question 14:

Let Z be the set of all integers and Z0 be the set of all non-zero integers. Let a relation R on Z × Z0 be defined as
(a, b) R (c, d) ⇔ ad = bc for all (a, b), (c, d) ∈ Z × Z0,
Prove that R is an equivalence relation on Z × Z0.

We observe the following properties of R.

Reflexivity:

Symmetry:

Transitivity:

#### Question 15:

If R and S are relations on a set A, then prove that
(i) R and S are symmetric ⇒ RS and RS are symmetric
(ii) R is reflexive and S is any relation ⇒ RS is reflexive.

(i) R and S are symmetric relations on the set A.

Also,

(ii) R is reflexive and S is any relation.

#### Question 16:

If R and S are transitive relations on a set A, then prove that RS may not be a transitive relation on A.

Let  A = {a, b, c} and R and S be two relations on A, given by

R = {(a, a), (a, b), (b, a), (b, b)} and
S = {(b, b), (b, c), (c, b), (c, c)}

Here, the relations R and S are transitive on A.

Hence, R$\cup$S is not a transitive relation on A.

#### Question 17:

Let C be the set of all complex numbers and Cbe the set of all no-zero complex numbers. Let a relation R on Cbe defined as

${z}_{1}$  is real for all C0.

Show that R is an equivalence relation.

(i) Test for reflexivity:

Since, $\frac{{z}_{1}-{z}_{1}}{{z}_{1}+{z}_{1}}=0$, which is a real number.

So,

Hence, R is relexive relation.

(ii) Test for symmetric:

Let .

Then, $\frac{{z}_{1}-{z}_{2}}{{z}_{1}+{z}_{2}}=x$, where x is real

So,

Hence, R is symmetric relation.

(iii) Test for transivity:

Let .

Then,

Also,

Dividing (1) and (2), we get

Hence, R is transitive relation.

From (i), (ii), and (iii),

R is an equivalenve relation.

#### Question 1:

Let R be a relation on the set N given by
R = {(a, b) : a = b − 2, b > 6}. Then,
(a) (2, 4) ∈ R
(b) (3, 8) ∈ R
(c) (6, 8) ∈ R
(d) (8, 7) ∈ R

(c) (6, 8) ∈ R

#### Question 2:

If a relation R is defined on the set Z of integers as follows:
(a, b) ∈ Ra2 + b2 = 25. Then, domain (R) is
(a) {3, 4, 5}
(b) {0, 3, 4, 5}
(c) {0, ± 3, ± 4, ± 5}
(d) none of these

(c) {0, ± 3, ± 4, ± 5}

#### Question 3:

R is a relation on the set Z of integers and it is given by
(x, y) ∈ R ⇔ | xy | ≤ 1. Then, R is
(a) reflexive and transitive
(b) reflexive and symmetric
(c) symmetric and transitive
(d) an equivalence relation

(b) reflexive and symmetric

#### Question 4:

The relation R defined on the set A = {1, 2, 3, 4, 5} by
R = {(a, b) : | a2b2 | < 16} 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

(d) none of these

R is given by {(1, 2), (2, 1), (2, 3), (3, 2), (3, 4), (4, 3), (4, 5), (5, 4), (1, 3), (3, 1), (1, 4), (4, 1) ,(2, 4), (4, 2)}, which is not mentioned in (a), (b) or (c).

#### Question 5:

Let R be the relation over the set of all straight lines in a plane such that  l1 R l2l 1l2. Then, R is
(a) symmetric
(b) reflexive
(c) transitive
(d) an equivalence relation

(a) symmetric

A = Set of all straight lines in the plane

#### Question 6:

If A = {a, b, c}, then the relation R = {(b, c)} on A is
(a) reflexive only
(b) symmetric only
(c) transitive only
(d) reflexive and transitive only

(c) transitive only

The relation R = {(b,c)} is neither reflexive nor symmetric because every element of A is not related to itself. Also, the ordered pair of R obtained by interchanging its elements is not contained in R.

We observe that R is transitive on A because there is only one pair.

#### Question 7:

Let A = {2, 3, 4, 5, ..., 17, 18}. Let '≃' be the equivalence relation on A × A, cartesian product of A with itself, defined by (a, b) ≃ (c, d) if ad = bc. Then, the number of ordered pairs of the equivalence class of (3, 2) is
(a) 4
(b) 5
(c) 6
(d) 7

(c) 6

The ordered pairs of the equivalence class of (3, 2) are {(3, 2), (6, 4), (9, 6), (12, 8), (15, 10), (18, 12)}.
We observe that these are 6 pairs.

#### 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

(a) 1

The required relation is R.
R = {(1, 2), (1, 3), (1, 1), (2, 2), (3, 3), (2, 1), (3, 1)}

Hence, there is only 1 such relation that is reflexive and symmetric, but not transitive.

#### Question 9:

The relation 'R' in N × N such that
(a, b) R (c, d) ⇔ a + d = b + c is
(a) reflexive but not symmetric
(b) reflexive and transitive but not symmetric
(c) an equivalence relation
(d) none of the these

(c) an equivalence relation

We observe the following properties of relation R.

Hence, R is an equivalence relation on N.

#### 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

(c) {1}

Here,

Thus,
Range of R = {1}

#### Question 11:

A relation R is defined from {2, 3, 4, 5} to {3, 6, 7, 10} by : x R yx 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}

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

The relation R is defined as

Hence, the domain of R includes all the values of x, i.e. {2, 3, 4, 5}.

#### Question 12:

A relation ϕ from C to R is defined by x ϕ y ⇔ | x | = y. Which one is correct?
(a) (2 + 3 i) ϕ 13
(b) 3 ϕ (−3)
(c) (1 + i) ϕ 2
(d) i ϕ 1

(d)  i ϕ 1

#### 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}

(c) {2,4,6}

The relation R is defined as

Domain of R is all values of x$\in$N satisfying the relation R. Also, there are only three values of x that result in y, which is a natural number. These are {2, 6, 4}.

#### Question 14:

R is a relation from {11, 12, 13} to {8, 10, 12} defined by y = x − 3. Then, R−1 is
(a) {(8, 11), (10, 13)}
(b) {(11, 8), (13, 10)}
(c) {(10, 13), (8, 11)}
(d) none of these

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

The relation R is defined by

#### 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) identify relation
(b) reflexive
(c) symmetric
(d) antisymmetric

(b) reflexive

#### Question 16:

Let A = {1, 2, 3} and B = {(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

(c)  transitive

#### Question 17:

If R is the largest equivalence relation on a set A and S is any relation on A, then
(a) RS
(b) SR
(c) R = S
(d) none of these

(b) SR

Since R is the largest equivalence relation on set A,

Since S is any relation on A,

So, SR

#### Question 18:

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

(d) none of these

The relation R is defined as

#### 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

(d) all the three options

Hence, R is an equivalence relation on A.

#### 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

(a) symmetric and transitive only

#### Question 21:

If A = {1, 2, 3}, then a relation R = {(2, 3)} on A is
(a) symmetric and transitive only
(b) symmetric only
(c) transitive only
(d) none of these

(c) transitive only

The relation R is not reflexive because every element of A is not related to itself. Also, R is not symmetric since on interchanging the elements, the ordered pair in R is not contained in it.

R is transitive by default because there is only one element in it.

#### 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 symmetric
(c) R is symmetric and transitive but not reflexive
(d) R is an equivalence relation

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

#### Question 23:

Let A = {1, 2, 3}. Then, the number of equivalence relations containing (1, 2) is
(a) 1
(b) 2
(c) 3
(d) 4

(b) 2

There are 2 equivalence relations containing {1, 2}.
R = {(1, 2)}
S = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2), (1, 3), (3, 1)}

#### 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

(c) an equivalence relation

Hence, R is an equivalence relation on A.

#### Question 25:

S is a relation over the set R of all real numbers and it is given by
(a, b) ∈ Sab ≥ 0. Then, S is
(a) symmetric and transitive only
(b) reflexive and symmetric only
(c) antisymmetric relation
(d) an equivalence relation

(d) an equivalence relation

Reflexivity: Let a$\in$R
Then,

So, S is reflexive on R.

Symmetry: Let (a, b)$\in$S
Then,

So, S is symmetric on R.

Transitivity:

Hence, S is an equivalence relation on R.

#### 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 xy
(b) x R y : if x = y
(c) x R y : if xy is an even integer
(d) x R y : if xy (mod 3)

(a) x R y : if xy

Clearly, R is not symmetric because x < y does not imply y < x.

Hence, (a) is not an equivalence relation.

#### Question 27:

Mark the correct alternative in the following question:

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

We have,

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

Hence, the correct alternative is option (a).

#### Question 28:

Mark the correct alternative in the following question:

The relation S defined on the set R of all real number by the rule aSb iff a $\ge$ b is

(a) an equivalence relation
(b) reflexive, transitive but not symmetric
(c) symmetric, transitive but not reflexive
(d) neither transitive nor reflexive but symmetric

We have,

S = {(a, b) : a $\ge$ b; a, b $\in$ R}

Hence, the correct alternative is option (b).

#### Question 29:

Mark the correct alternative in the following question:

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

(a) 1                                  (b) 2                                  (c) 3                                  (d) 5

Hence, the correct alternative is option (d).

#### Question 30:

Mark the correct alternative in the following question:

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

(a) Reflexive and symmetric                                                         (b) Transitive and symmetric
(c) Equivalence                                                                          (d) Reflexive, transitive but not symmetric                  [NCERT EXEMPLAR]

We have,

R = {(m, n) : n divides m; m, n $\in$ N}

Hence, the correct alternative is option (d).

#### Question 31:

Mark the correct alternative in the following question:

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

(a) reflexive                                 (b) symmetric                                  (c) transitive                                  (d) none of these
[NCERT EXEMPLAR]

Hence, the correct alternative is option (b).

#### Question 32:

Mark the correct alternative in the following question:

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

a) reflexive but not symmetric                                                                               (b) transitive but not symmetric
c) equivalence                                                                                                     (d) none of these

Hence, the correct alternative is option (c).

#### Question 33:

Mark the correct alternative in the following question:

Consider a non-empty set consisting of children in a family and a relation R defined as aRb 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 and transitive

Hence, the correct alternative is option (b).

#### Question 34:

Mark the correct alternative in the following question:

For real numbers x and y, define xRy iff $x-y+\sqrt{2}$ is an irrational number. Then the relation R is

(a) reflexive                          (b) symmetric                          (c) transitive                          (d) none of these

Hence, the correct alternative is option (a).

#### Question 35:

If a relation R on the set (1, 2, 3) be defined by = {(1, 2)}, then R is
(a) reflexive           (b) transitive               (c) symmetric             (d) none of these

Given: A relation R on the set {1, 2, 3} be defined by = {(1, 2)}.

= {(1, 2)}

Since, (1, 1) ∉ R
Therefore, It is not reflexive.

Since, (1, 2) ∈ R but (2, 1) ∉ R
Therefore, It is not symmetric.

But there is no counter example to disapprove transitive condition.
Therefore, it is transitive.

Hence, the correct option is (b).

#### Question 1:

If R is a relation in Z, then the domain of R is ______________________.

Given: R =

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

Therefore, Domain of R = {−2, −1, 0, 1, 2}

Hence, if R =  is a relation in Z, then the domain of R is {−2, −1, 0, 1, 2}.

#### Question 2:

Let R be a relation in N defined by R ={(x, y): + 2y = 8}, then the range of R is ___________________.

Given: R = {(x, y): + 2y = 8} where x, y ∈ N

R = {(6, 1), (4, 2), (2, 3)}

Therefore, Range of R = {1, 2, 3}

Hence, the range of R is {1, 2, 3}.

#### Question 3:

The number of relations on a finite set having 5 elements is __________________.

Let R be a relation on A, where A contains 5 elements.

R is a subset of A × A.

Number of elements in A × A = 5 × 5 = 25

Number of relations = Number of subsets of A × A = 225

Hence, the number of relations on a finite set having 5 elements is 225.

#### Question 4:

Let A = {1, 2, 3, 4} and R be the relation on A defined by {(a, b): a, b ∈ A, a×b is an even number}, then the range of R is __________________.

Given: R = {(a, b): a, b ∈ A, a × b is an even number}, where A = {1, 2, 3, 4}.

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

Therefore, Range of R = {1, 2, 3, 4}

Hence, the range of R is {1, 2, 3, 4}.

#### Question 5:

Let A = {1, 2, 3, 4, 5} The domain of the relation on A defined by R ={(x,y): y = 2x-1},is__________________.

Given: R = {(x, y): y = 2x − 1}, where A = {1, 2, 3, 4, 5} and x, y ∈ A.

R = {(1, 1), (2, 3), (3, 5)}

Therefore, Domain of R = {1, 2, 3}.

Hence, the domain of the relation on A defined by R = {(x, y): y = 2x − 1}, is {1, 2, 3}.

#### Question 6:

If R s a relation defined on set A ={1, 2, 3} by the rule (a,b) $\in R⇔\left|{a}^{2}-{b}^{2}\right|\le 5,$ then R-1 =____________________.

Given: R = {(a, b): $\left|{a}^{2}-{b}^{2}\right|\le 5$}, where A = {1, 2, 3} and ab ∈ A.

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

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

Hence, if R is a relation defined on set A = {1, 2, 3} by the rule (a,b$\in R⇔\left|{a}^{2}-{b}^{2}\right|\le 5,$ then R-1 = R.

#### Question 7:

If R is a relation from A = {11, 12, 13} to B = {8, 10 12} defined by y = x-3, then R-1 =_______________________.

Given: R = {(x, y): y = x − 3, xA and yB}, where A = {11, 12, 13} and B = {8, 10 12}.

R = {(11, 8), (13, 10)}

Therefore, R= {(8, 11), (10, 13)}

Hence, R-1 {(8, 11), (10, 13)}.

#### Question 8:

The smallest equivalence relation on the set A = {a, b, c, d} is _________________________.

Given: A = {a, b, c, d}

Identity relation is the smallest equivalence relation.

Therefore, R = {(a, a), (bb), (cc)} is the smallest equivalence relation.

Hence, the smallest equivalence relation on the set A = {a, b, c, d} is {(aa), (bb), (cc)}.

#### Question 9:

The largest equivalence relation on the set A = {1, 2, 3} is ___________________.

Given: A = {1, 2, 3}

The largest equivalence relation contains all the possible ordered pairs.

Therefore, R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2), (1, 3), (3, 1)} is the largest equivalence relation.

Hence, the largest equivalence relation on the set A = {1, 2, 3} is {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2), (1, 3), (3, 1)}.

#### Question 10:

Let R be the equivalence relation on the set Z of integers given by R = {(a, b): 3 divides a-b}. Then the equivalence class [0] is equal to ____________________.

Given: R is the equivalence relation on the set Z of integers given by R = {(a, b): 3 divides a − b}.

To find the equivalence class [0], we put b = 0 in the given relation and find all the possible values of a.

Thus,
R = {(a, 0): 3 divides a − 0}
⇒ a − 0 is a multiple of 3
a is a multiple of 3
⇒ a = 3n , where n ∈ Z
⇒ a = 0, ±3, ±6, ±9, ....

Therefore, equivalence class [0] = {0, ±3, ±6, ±9, ....}

Hence, the equivalence class [0] is equal to {0, ±3, ±6, ±9, ....}.

#### Question 11:

Let R be a relation on the set Z of all integers defined as (x, y) ∈ R ⇔ x-y is divisible by 2. Then, the equivalence class [1] is _________________.

Given: R is the equivalence relation on the set Z of integers defined as (x, y) ∈ R ⇔ x − y is divisible by 2.

To find the equivalence class [1], we put y = 1 in the given relation and find all the possible values of x.

Thus,
R = {(x, 1): x − 1 is divisible by 2}
⇒ x − 1 is divisible by 2
⇒ x = ±1, ±3, ±6, ±9, ....

Therefore, equivalence class [0] = {±1, ±3, ±6, ±9, ....}

Hence, the equivalence class [1] is {±1, ±3, ±6, ±9, ....}.

#### Question 12:

The relation R = {(1, 2,), (1, 3)} on set A = [1, 2, 3] is _________________ only.

Given: A relation R on the set {1, 2, 3} be defined by = R = {(1, 2,), (1, 3)}.

R = {(1, 2,), (1, 3)}

Since, (1, 1) ∉ R
Therefore, It is not reflexive.

Since, (1, 2) ∈ but (2, 1) ∉ R
Therefore, It is not symmetric.

But there is no counter example to disapprove transitive condition.
Therefore, it is transitive.

Hence, The relation R = {(1, 2,), (1, 3)} on set A = {1, 2, 3} is transitive only.

#### Question 1:

Write the domain of the relation R defined on the set Z of integers as follows:
(a, b) ∈ Ra2 + b2 = 25

Domain of R is the set of values satisfying the relation R.
As a should be an integer, we get the given values of a:

#### Question 2:

If R = {(x, y) : x2 + y2 ≤ 4; x, yZ} is a relation on Z, write the domain of R.

Domain of R is the set of values of x satisfying the relation R.
As x must be an integer, we get the given values of x:

#### Question 3:

Write the identity relation on set A = {a, b, c}.

Identity set of A is
I = {(a, a), (b, b), (c, c)}

Every element of this relation is related to itself.

#### Question 4:

Write the smallest reflexive relation on set A = {1, 2, 3, 4}.

Here,
A
= {1, 2, 3, 4}
Also, a relation is reflexive iff every element of the set is related to itself.

So, the smallest reflexive relation on the set A is
R = {(1, 1), (2, 2), (3, 3), (4, 4)}

#### Question 5:

If R = {(x, y) : x + 2y = 8} is a relation on N by, then write the range of R.

R = {(x, y) : x + 2y = 8, x, y$\in$N}
Then, the values of y can be 1, 2, 3 only.
Also, y = 4 cannot result in x = 0 because x is a natural number.

Therefore, range of R is {1, 2, 3}.

#### Question 6:

If R is a symmetric relation on a set A, then write a relation between R and R−1.

Here, R is symmetric on the set A.

#### Question 7:

Let R = {(x, y) : |x2y2| <1) be a relation on set A = {1, 2, 3, 4, 5}. Write R as a set of ordered pairs.

R is the set of ordered pairs satisfying the above relation. Also, no two different elements can satisfy the relation; only the same elements can satisfy the given relation.

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

#### Question 8:

If A = {2, 3, 4}, B = {1, 3, 7} and R = {(x, y) : xA, yB and x < y} is a relation from A to B, then write R−1.

Since R = {(x, y) : x $\in$ A, y $\in$ A and x < y},
R = {(2, 3), (2, 7), (3, 7), (4, 7)}

So, R-1 = {(3, 2), (7, 2), (7, 3), (7, 4)}

#### Question 9:

Let A = {3, 5, 7}, B = {2, 6, 10} and R be a relation from A to B defined by R = {(x, y) : x and y are relatively prime}. Then, write R and R−1.

R = {(x, y) : x and y are relatively prime}
Then,

R = {(3, 2), (5, 2), (7, 2), (3, 10), (7, 10), (5, 6), (7, 6)}

So, R-1 = {(2, 3), (2, 5), (2, 7), (10, 3), (10, 7), (6, 5), (6, 7)}

#### Question 10:

Define a reflexive relation.

A relation R on A is said to be reflexive iff every element of A is related to itself.

i.e. R is reflexive

#### Question 11:

Define a symmetric relation.

A relation R on a set A is said to be symmetric iff

#### Question 12:

Define a transitive relation.

A relation R on a set A is said to be transitive iff

#### Question 13:

Define an equivalence relation.

A relation R on set A is said to be an equivalence relation iff
(i) it is reflexive,
(ii) it is symmetric and
(iii) it is transitive.

Relation R on set A satisfying all the above three properties is an equivalence relation.

#### Question 14:

If A = {3, 5, 7} and B = {2, 4, 9} and R is a relation given by "is less than", write R as a set ordered pairs.

#### Question 15:

A = {1, 2, 3, 4, 5, 6, 7, 8} and if R = {(x, y) : y is one half of x; x, yA} is a relation on A, then write R as a set of ordered pairs.

Since R = {(x, y) : y is one half of x; x, y$\in$A}

So, R = {(2, 1), (4, 2), (6, 3), (8, 4)}

#### Question 16:

Let A = {2, 3, 4, 5} and B = {1, 3, 4}. If R is the relation from A to B given by a R b if "a is a divisor of b". Write R as a set of ordered pairs.

So, R = {(2, 4), (3, 3), (4, 4)}

#### Question 17:

State the reason for the relation R on the set {1, 2, 3} given by R = {(1, 2), (2, 1)} to be transitive.

#### Question 18:

Let R = {(aa3) : a is a prime number less than 5} be a relation. Find the range of R.                                                                    [CBSE 2014]

We have,
R = {(aa3) : a is a prime number less than 5}
Or,
R = {(2, 8), (3, 27)}

So, the range of R is {8, 27}.

#### Question 19:

Let R be the equivalence relation on the set Z of the integers given by R = {(a, b) : 2 divides a $-$ b}. Write the equivalence class [0].
[NCERT EXEMPLAR]

We have,
An equivalence relation, R = {(ab) : 2 divides a $-$ b}

#### Question 20:

For the set A = {1, 2, 3}, define 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.

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

As, (a, a$\in$ R, for all values of a $\in$ A
So, R is a reflexive relation

R can be a symmetric and transitive relation only when element (3, 1) is added

Hence, the ordered pairs to be added to R to make the smallest equivalence relation is (3, 1).

#### 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?

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

#### 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 as a set of ordered pairs.

As, R = {(ab) : |a2 $-$ b2| < 8}
So, 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 defined on N by aRb iff 2a + 3b = 30. Then write R as a set of ordered pairs.

As, R = {(a, b) : 2a + 3b = 30; a, b $\in$ N}

So, R = {(3, 8), (6, 6), (9, 4), (12, 2)}

#### Question 24:

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