Question

If $R=\left\{(3,3),(6,6),(9,9),(12,12),(6,12),(3,9),(3,12)(3,6)\right\}$ is a relation on the set $A=\left\{3,6,9,12\right\}$. Then, the relation is:

• A. Reflexive and symmetric
• B. An equivalence relation
• C. Reflexive only
• D. Reflexive and transitive

Answer 1

The correct option is C

Reflexive only

Explanation for correct options.

Option C: reflexive only

$R=\left\{(3,3),(6,6),(9,9),(12,12),(6,12),(3,9),(3,12)(3,6)\right\}$ and $A=\left\{3,6,9,12\right\}$.

Clearly, every element of $A$is related to itself hence it is reflexive.

Hence, option C is correct.

Explanation for incorrect options.

A relation is said to be reflexive if every element is related to itself.

A relation is said to be symmetric if $xRy$ then $yRx$.

A relation is said to be transitive if $xRy,yRz$ then $xRz$.

Option A: reflexive and symmetric

Given that, $R=\left\{(3,3),(6,6),(9,9),(12,12),(6,12),(3,9),(3,12)(3,6)\right\}$and $A=\left\{3,6,9,12\right\}$

Clearly, every element of $A$is related to itself hence it is reflexive.

It is not symmetric as $\left(6,12\right)\in R$ but $\left(12,6\right)\notin R$.

Hence, option A is incorrect.

Option B: an equivalence relation

Hence, option B is incorrect.

Option D: reflexive and transitive

Relation $R$is not transitive as the property $xRy,yRz$ then $xRz$ is not satisfied.

Hence, option D is incorrect.

Hence, the correct answer is an option (C).