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
(1) reflexive and symmetric
(2) an equivalence relation
(3) reflexive only
(4) Reflexive and Transitive.

Answer 1

For solving this question you should know about the reflexive and equivalence relation. In this problem we will check the sets according to the definition of Reflexive and equivalence. And then verify these with the sets. If these are verifying then the relation will be there unless no relation will be there.

Complete step by step answer:
According to the question it is asked to us to find the relations type of two given sets.
As we know that in terms of relations, the reflexive can be defined as $\left( a,a \right)\in R\forall a\in X$ or as $I\subseteq R$ where I is the identity relation on A. Thus, it has a reflexive property.
When every real number is equal to itself the relation “is equal to” is used on the set of real numbers. A reflexive relation is said to have the reflexive property.
So, if we see to the question, then the sets are given as:
$R=\left[ \left( 3,3 \right),\left( 6,6 \right),\left( 9,9 \right),\left( 12,12 \right),\left( 6,12 \right),\left( 3,9 \right),\left( 3,12 \right),\left( 3,6 \right) \right]$ be a relation on the set A=\left\\{ 3,6,9,12 \right\\}
So, if we cross check it then,
(d) : for $\left( 3,9 \right)\in R$, $\left( 9,3 \right)\notin R$
Therefore the relation is not symmetric.
(a) and (b) are out of court because we need to prove reflexivity and transitivity.
For reflexivity $a\in R,\left( a,a \right)\in R$ which is hold i.e., R is reflexive, Again,
For transitivity of $\left( a,b \right)\in R,\left( b,c \right)\in R$
$\Rightarrow \left( a,c \right)\in R$
which is also true in R.

So, the correct answer is “Option 3”.

Note: While solving these types of questions you have to check them properly with every element of the set and with correct definition of the reflexive and transitivity and symmetricity. Because if any single element leaves by mistake then it can make our whole question wrong.