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Question

Let $R$ be a relation on the set $N$ be defined by $open curly brackets open parentheses x comma y close parentheses vertical line x comma y element of N comma 2 blank x plus y equals 41 close curly brackets.$ Then, $R$ is

Reflexive
Symmetric
Transitive
None of these

The correct answer is: None of these

We have
$R equals left curly bracket open parentheses 139 close parentheses comma open parentheses 237 close parentheses comma open parentheses 335 close parentheses comma open parentheses 433 close parentheses comma open parentheses 531 close parentheses comma open parentheses 629 close parentheses comma$
$open parentheses 727 close parentheses comma open parentheses 825 close parentheses comma open parentheses 923 close parentheses comma open parentheses 1021 close parentheses comma open parentheses 1119 close parentheses comma open parentheses 1217 close parentheses comma$
$open parentheses 1315 close parentheses comma open parentheses 1413 close parentheses comma open parentheses 1511 close parentheses comma open parentheses 169 close parentheses comma open parentheses 177 close parentheses comma open parentheses 185 close parentheses comma$
$open parentheses 193 close parentheses comma left parenthesis 201 right parenthesis right curly bracket$
Since $open parentheses 139 close parentheses element of R comma$ but $left parenthesis 391 right parenthesis not an element of R$
Therefore, $R$ is not symmetric
Clearly, $R$ is not reflexive. Now, $left parenthesis 1511 right parenthesis element of R$ and $left parenthesis 1119 right parenthesis element of R$ but $left parenthesis 1519 right parenthesis not an element of R$
So, $R$ is not transitive

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