Given the relation R = {(1, 2), (2, 3)} on the set A = {1, 2, 3}, the

1.	Given the relation R = {(1, 2), (2, 3)} on the set A = {1, 2, 3}, the minimum number of ordered pairs which can be added to R to make it an equivalence relation is(a) 2 (b) 3 (c) 5 (d) 7
Answer» (d) 7 A = {1, 2, 3}. R = {(1, 2), (2, 3)} To make R an equivalence relation, it should be: (i) Reflexive: So three more ordered pairs (1, 1), (2, 2), (3, 3) should be added to R to make it reflexive. (ii) Symmetric : As R contains (1, 2) and (2, 3) so two more ordered pairs (2, 1) and (3, 2) should be added to make it symmetric. (iii) Transitive: (1, 2) ∈ R, (2, 3) ∈ R. So to make R transitive (1, 3) should be added to R. Also to maintain the symmetric property (3, 1) should then be added to R. So, R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (2, 1), (3, 2), (1, 3), (3, 1)} is an equivalence relation. So minimum 7 ordered pairs are to be added.

Given the relation R = {(1, 2), (2, 3)} on the set A = {1, 2, 3}, the minimum number of ordered pairs which can be added to R to make it an equivalence relation is(a) 2 (b) 3 (c) 5 (d) 7

Answer»

(d) 7

A = {1, 2, 3}.

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

To make R an equivalence relation, it should be:

(i) Reflexive: So three more ordered pairs (1, 1), (2, 2), (3, 3) should be added to R to make it reflexive.

(ii) Symmetric : As R contains (1, 2) and (2, 3) so two more ordered pairs (2, 1) and (3, 2) should be added to make it symmetric.

(iii) Transitive: (1, 2) ∈ R, (2, 3) ∈ R. So to make R transitive (1, 3) should be added to R. Also to maintain the symmetric property (3, 1) should then be added to R.

So, R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (2, 1), (3, 2), (1, 3), (3, 1)} is an equivalence relation. So minimum 7 ordered pairs are to be added.

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