Question

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

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

Answer 1

The correct option is C

Reflexive and transitive

Explanation for the correct options:

Step 1: Check for reflexive relation.

For a relation R in a set A,

A relation is reflexive if $(a,a)\in R$ for every $a\in A$.

Here, $\{(3,3),(6,6),(9,9),(12,12)\}\in R$ for every $\{3,6,9,12\}\in A$,

Therefore, the relation is reflexive.

Step 2: Check for symmetric relation.

For a relation R in a set A,

A relation is symmetric if $(a,b)\in R$ then $(b,a)\in R$.

Here, $\left\{\right(6,12),(3,9),(3,12),(3,6)\}\in R$ but $\left\{\right(12,6),(9,3),(12,3),(6,3)\}\notin R$,

Therefore, the relation is not symmetric.

Step 3: Check for transitive relation.

For a relation R in a set A,

A relation is transitive if $(a,b)\in R$ and $(b,c)\in R$ then $(a,c)\in R$.

Here, $\left\{\right(3,6),(6,12\left)\right\}\in R$ and $(3,12)\in R$, similarly for pairs such $\left\{\right(3,9),(9,9\left)\right\}\in R$ also $(3,9)\in R$

Therefore, the relation is transitive.

We can conclude that the given relation is reflexive and transitive but not symmetric.

Hence, the correct option is (C).