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Easy

Question

If $A equals left curly bracket 4 comma blank 6 comma blank 10 comma blank 12 right curly bracket$ and $R$ is a relation defined on $A$ as “two elements are related iff they have exactly one common factor other than 1”. Then the relation $R$ is

Antisymmetric
Only transitive
Only symmetric
Equivalence

The correct answer is: Only symmetric

Clearly,
$R equals left curly bracket open parentheses 46 close parentheses comma open parentheses 410 close parentheses comma open parentheses 64 close parentheses comma open parentheses 104 close parentheses open parentheses 610 close parentheses comma open parentheses 106 close parentheses comma open parentheses 1012 close parentheses comma left parenthesis 1210 right parenthesis right curly bracket$
Clearly, $R$ is symmetric
$left parenthesis 610 right parenthesis element of R$ and $left parenthesis 1012 right parenthesis element of R$ but $left parenthesis 612 right parenthesis not an element of R$
So, $R$ is not transitive
Also, $R$ is not reflexive

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