# Mark the tick against the correct answer in the following:

Question:

Mark the tick against the correct answer in the following:

Let $S$ be the set of all straight lines in a plane. Let $R$ be a relation on $S$ defined by a $R$ b $\Leftrightarrow$ a $\|$ b. Then, $R$ is

A. reflexive and symmetric but not transitive

B. reflexive and transitive but not symmetric

C. symmetric and transitive but not reflexive

D. an equivalence relation

Solution:

According to the question ,

Given set $S=\{x, y, z\}$

And $R=\{(x, x),(y, y),(z, z)\}$

Formula

For a relation $R$ in set $A$

Reflexive

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

Symmetric

The relation is Symmetric if $(a, b) \in R$, then $(b, a) \in R$

Transitive

Relation is Transitive if $(a, b) \in R \&(b, c) \in R$, then $(a, c) \in R$

Equivalence

If the relation is reflexive, symmetric and transitive, it is an equivalence relation.

Check for reflexive

Since, $(x, x) \in R,(y, y) \in R,(z, z) \in R$

Therefore, $R$ is reflexive ....... (1)

Check for symmetric

Since,$(x, x) \in R$ and $(x, x) \in R$

$(y, y) \in R$ and $(y, y) \in R$

$(z, z) \in R$ and $(z, z) \in R$

Therefore, $\mathrm{R}$ is symmetric ....... (2)

Check for transitive

Here, $(x, x) \in R$ and $(y, y) \in R$ and $(z, z) \in R$

Therefore, $\mathrm{R}$ is transitive ....... (3)

Now, according to the equations (1), (2), (3)

Correct option will be (D)