Question

The relation $\text{'}\text{'}$is parallel to $\text{'}\text{'}$ on a set $S$ of all straight lines in a plane is :

• A. Symmetric only
• B. Reflexive and Transitive only
• C. Transitive only
• D. An equivalence relation

Answer 1

The correct option is D

An equivalence relation

Explanation for the correct answer:

Option $\left(D\right)$: An equivalence relation

Let $R=\left\{\left(x,y\right):\text{line}x\text{is parallel to line}y,xy\in \text{set of coplaner staright lines}\right\}$

Where every lines is parallel to itself.

A relation $R$ on set $A$is said to be Reflexive. if $\left(a,a\right)\in R,a\in A$

If $x\in S$, then $\left(x,x\right)\in R$

Thus, $R$ is Reflexive.

A relation $R$ on set $A$is said to be Symmetric, if $\left(a,b\right)\in R\left(b,a\right)\in R$

If $\left(x,y\right)\in R$

$\Rightarrow x\parallel y\Rightarrow y\parallel x$

$\Rightarrow \left(y,x\right)\in R$

Thus, $R$ is Symmetric.

A relation $R$ on set $A$is said to be transitive if $\left(a,b\right)\in R\left(b,c\right)\in R\left(a,c\right)\in R$

If $x\parallel y$ and $y\parallel z$

$\Rightarrow x\parallel z$

$\Rightarrow \left(x,z\right)\in R$

Thus, $R$ is transitive.

Since, $R$ is Reflexive, Symmetric and transitive, which is an equivalence Relations.

Therefore, Option $\left(D\right)$ is the correct answer.