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Question

The relation $R equals open curly brackets open parentheses 13 close parentheses comma left parenthesis 35 right parenthesis close curly brackets$ is defined on the set with minimum number of elements of natural numbers. The minimum number of elements to be included in $R$ so that $R$ is an equivalence relation, is

(a) 5
6
7
8

The correct answer is: (a) 5

To make $R$ a reflexive relation, we must have $open parentheses 11 close parentheses comma open parentheses 33 close parentheses$ and $open parentheses 55 close parentheses$ in it. In order to make $R$ a symmetric relation, we must inside $left parenthesis 31 right parenthesis$ and $left parenthesis 53 right parenthesis$ in it.
Now, $open parentheses 13 close parentheses element of R$ and $open parentheses 35 close parentheses element of R.$ So, to make $R$ a transitive relation, we must have, $open parentheses 15 close parentheses element of R.$ But, $R$ must be symmetric also. So, it should also contain $open parentheses 51 close parentheses.$ Thus, we have
$R equals left curly bracket open parentheses 11 close parentheses comma open parentheses 33 close parentheses comma open parentheses 55 close parentheses comma open parentheses 13 close parentheses comma open parentheses 35 close parentheses comma open parentheses 31 close parentheses comma open parentheses 53 close parentheses comma open parentheses 15 close parentheses comma left parenthesis 51 right parenthesis right curly bracket$
Clearly, it is an equivalence relation on $A open curly brackets 13 comma 5 close curly brackets$

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