Question

A relation R is defined on the set A=$\left\{ {1,2,3,4,5,6} \right\}$ by R=$\left\{ {\left( {x,y} \right):y\;is\;divisible\;by\;x} \right\}$ .
Verify whether R is symmetric and reflexive or not. Give reason:

Answer 1

Hint:We should have knowledge of topic Relations to solve this problem. First of all , recall the definitions & conditions of being reflexive & symmetric & then focus on the given set A & condition mentioned for R set . Now , check whether all elements of A are satisfying conditions of being reflexive to R or not & also prove it by giving examples by taking elements from given non-empty set A .

Complete step-by-step answer:
We know , a relation R is said to be reflexive on set A if every element of A is related to itself ( where A is a non empty set) . i.e , if \[\left( {a,a} \right)\]$ \in R$ for every $a \in A$.
So , clearly a relation R on A is reflexive if there exists an element $a$\[ \in A\] such that \[\left( {a,a} \right)\]$ \notin R$
Given, $A = \left\{ {1,2,3,4,5,6} \right\}$ thus A is a non-empty set.
According to question, $R = \left\{ {\left( {x,y} \right):y\;is\;divisible\;by\;x} \right\}$
We know that any number $\left( x \right)$ is divisible by itself for all \[x \in A\].
$\therefore \left( {x,x} \right) \in R$ Thus, R satisfies the conditions of being a reflexive on A.
 Hence R is reflexive on A.

Now, by definition, A symmetric relation is a type of binary relation. A binary relation R over a set X is symmetric if ${R^T}$represents the converse of R, then R is symmetric if and only if $R = {R^T}$. i.e, if \[\left( {a,b} \right)\; \in A\]$ \in R$ $ \Rightarrow (b,a) \in R$ for all \[\left( {a,b} \right)\; \in A\][ A being a given non empty set ]
Now, $\left( {2,4} \right)$$ \in R$ $\left[ {\because 4\;is\;divisible\;by\;2} \right]$
But, $\left( {4,2} \right)$$ \notin R$ $\left[ {\because 2\;is\;not\;divisible\;by\;4} \right]$
$\therefore \;R\;is\;not\;symmetric.$
Hence, R is reflexive on A but not symmetric on A.

Note:To verify whether a given set is reflexive or not & symmetric or not, we should know the conditions to be reflexive or symmetric first. Then considering given sets we can apply the conditions so that along with reasons . Do each step carefully so that you can give reason for all steps . Definition & condition of reflexive & symmetric relation is most important to be remembered & accordingly applied.