Question

Let $R = \left\{ {\left( {3,3} \right)\left( {6,6} \right)\left( {9,9} \right)\left( {12,12} \right)\left( {6,12} \right)\left( {3,9} \right)\left( {3,12} \right)\left( {3,6} \right)} \right\}$ be a relation on the set $A = \left\{ {3,6,9,12} \right\}$. Then relation is
A. Reflexive only
B. Reflexive and transitive only
C. Reflexive and symmetric only
D. An equivalence relation

Answer 1

Hint: In this problem we have given a relation which is defined on the set $A$ and this set $A$ consists of some numbers. Now we are asked to find the name of the given relation. Also by using the given relation we need to find the relation reflexive, transitive or symmetry. If these are all satisfied then we can say the given relation is equivalence relation.

Formula used: Relation is reflexive. If $\left( {a,a} \right) \in R$ for every $a \in A$
Relation is symmetric. If $\left( {a,b} \right) \in R$, then $\left( {b,a} \right) \in R$
Relation is transitive. If $\left( {a,b} \right) \in R\& \left( {b,c} \right) \in R$, then $\left( {a,c} \right) \in R$
Suppose the relation is reflexive, symmetry and transitive then it is an equivalence relation.

Complete step-by-step solution:
Let $R = \left\{ {\left( {3,3} \right)\left( {6,6} \right)\left( {9,9} \right)\left( {12,12} \right)\left( {6,12} \right)\left( {3,9} \right)\left( {3,12} \right)\left( {3,6} \right)} \right\}$ be the given relation on the set $A = \left\{ {3,6,9,12} \right\}$.
Given relation R is reflexive:
For,$\left( {3,3} \right) \in R$ for every $3 \in A$. Similarly $\left( {6,6} \right)\left( {9,9} \right)\left( {12,12} \right) \in R$ for every$6,9,12 \in A$
Therefore the given relation $R$ on the set $A$ is reflexive.
Given relation R is not symmetry:
For,$\left( {6,12} \right) \in R$ but $\left( {12,6} \right) \notin R$ therefore the given relation is not symmetry.
Given relation is transitive:
For,$\left( {3,6} \right) \in R$ and $\left( {6,12} \right) \in R$ then $\left( {3,12} \right) \in R$. Therefore the given relation is transitive.
$\therefore $ The given relation is reflexive and transitive only.

Hence the answer is option (B).

Note: The reflexive property states that any real number, $a$ is equal to itself. That is $a = a$. An example for the reflexive property is a mirror. If you look at your reflection in a mirror, you can see yourself. The symmetric property states that for any real numbers $a$ and b, if $a = b$ then $b = a$. The transitive property states that for any real numbers $a$, $b$ and $c$, if $a = b$ and $b = c$ then $a = c$.