Page 14 - Mathematics Class - XII

Viva-Voce

1. What is a trivial relation?
Ans. Empty relation and universal relation are sometimes called trivial relation.
2. Let T be the set of all triangles in a plane with a relation R in T given by
R = {(T , T ) : T is congruent to T }. Is R an equivalence relation?
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Ans. R is reflexive: as for all triangles, T is congruent to T .
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R is symmetric: as T is congruent to T ⇒ T is congruent to T .
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R is transitive also, as T is congruent to T and T is congruent to T ⇒ T is congruent to T .
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Hence, R = {(T , T ) : T is congruent to T } is an equivalence relation.
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3. Let A = {a, b, c} and R = {(b, c), (a, b), (c, c)}, then identify the required minimum ordered pairs to make
the relation equivalence.
Ans. To make R reflexive: add (a, a) and (b, b)
To make R symmetric: add (b, a), (c, b)
To make R transitive: add (a, c)
To make R an equivalence relation: Now R is {(a, a), (b, b), (c, c), (a, b), (b, c), (b, a), (c, b) and (a, c)}.
4. What is an identity relation?
Ans. For a given set A, relation I = {(a, a), a ∈ A} is called the identity relation in A. In identity relation, every
element of A is related to itself only.
For example: If A = {2, 3, 4} then I = {(2, 2), (3, 3), (4, 4)} is the identity relation in A.
5. Is identity relation an equivalence relation?
Ans. Yes, identity relation is an equivalence relation.

MCQs

1. Let R be the relation in the set {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}.
Choose the correct option.

a) R is an equivalence relation. b) R is reflexive and transitive but not symmetric.
c) R is symmetric and transitive but not reflexive.
d) R is reflexive and symmetric but not transitive.
2. Let R be the relation in the set N given by R = {(a, b) : a = b – 2, b > 6}. Choose the correct answer.

a) (2, 4) ∈ R b) (3, 8) ∈ R c) (6, 8) ∈ R d) (8, 7) ∈ R
3. Let A = {1, 2, 3}, then number of equivalence relations containing (1, 2) is

a) 1 b) 2 c) 3 d) 4
4. Let A = {1, 2, 3}, then 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

Answers: 1. b) R is reflexive and transitive but not symmetric. 2. c) (6, 8) ∈ R
3. b) 2 4. a) 1

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