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Maths-

General

Easy

Question

Given the relation $R equals left curly bracket open parentheses 12 close parentheses comma left parenthesis 23 right parenthesis right curly bracket$ on the set $A equals open curly brackets 12 comma 3 close curly brackets comma$ the minimum number of ordered pairs which when added to $R$ make it an equivalence relation is

(a) 5
6
7
8

The correct answer is: 7

$R$ is reflexive if it contains $open parentheses 11 close parentheses comma open parentheses 22 close parentheses comma left parenthesis 33 right parenthesis$
$because open parentheses 12 close parentheses element of R comma left parenthesis 23 right parenthesis element of R$
$because R$ is symmetric, if $open parentheses 21 close parentheses comma left parenthesis 32 right parenthesis element of R$
Now, $R equals left curly bracket open parentheses 11 close parentheses comma open parentheses 22 close parentheses comma open parentheses 33 close parentheses comma open parentheses 21 close parentheses comma open parentheses 32 close parentheses comma open parentheses 23 close parentheses comma left parenthesis 12 right parenthesis right curly bracket$
$R$ will be transitive, if $open parentheses 31 close parentheses comma left parenthesis 13 right parenthesis element of R$
Thus, $R$ becomes an equivalence relation by adding $open parentheses 11 close parentheses blank open parentheses 22 close parentheses blank open parentheses 33 close parentheses comma open parentheses 21 close parentheses blank open parentheses 32 close parentheses comma open parentheses 13 close parentheses comma open parentheses 31 close parentheses.$ Hence, the total number of ordered pairs is 7

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