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 $R \quad b \Leftrightarrow a \perp b$. Then, $R$ is

A. reflexive but neither symmetric nor transitive

B. symmetric but neither reflexive nor transitive

C. transitive but neither reflexive nor symmetric

D. an equivalence relation

Solution:

According to the question,

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

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

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

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

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

Therefore, $R$ is symmetric ....... (2)

Check for transitive

Here, $(x, y) \in R$ and $(y, x) \in R$ but $(x, x) \notin R$

Therefore, $R$ is not transitive ....... (3)

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

Correct option will be (B)

 

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