Question

If $R$ is a relation defined as $aRb$, if $\left|a-b\right|>0$, then the relation is

• A. Reflexive
• B. Symmetric
• C. Transitive
• D. Symmetric and transitive

Answer 1

The correct option is D

Symmetric and transitive

Checking for the relation:

Step 1: Check for reflexive,

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

Assume any arbitrary element $$a$$, then $$\left|a-a\right|=0$$, which shows $a\notin R$.

Therefore, the relation $aRb$ is not reflexive.

Step 2: Check for symmetric:

A relation is said to be symmetric if $\left(a,b\right)\in R$ then $\left(b,a\right)\in R$

Assume that $$a$$ and $$b$$ be two different elements, then $$(a,b)\in R$$ which means $$\left|a-b\right|>0$$.

Since $\begin{array}{rcl}\left|a-b\right|& =& \left|b-a\right|\end{array}$, then we have

$\begin{array}{rcl}\left|b-a\right|& >& 0\end{array}$

Thus, $(b,a)\in R$ , then the relation $aRb$ is symmetric.

Step 3: Check for transitive:

A relation is said to be symmetric if $\left(a,b\right)\in R$ and $\left(b,c\right)\in R$ then $\left(a,c\right)\in R$

Assume that $\left(a,b\right)\in R$ and $b,c\in R$ which means $$\left|a-b\right|>0$$ and $$\left|b-c\right|>0$$

Add both the inequalities and we get

$\begin{array}{rcl}\left|a-c\right|& >& 0\end{array}$

Thus, $\left(a,c\right)\in R$ , then the relation is transitive.

Hence, the correct option is (D).