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
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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