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Mark the tick against the correct answer in the following:

Question:

Mark the tick against the correct answer in the following:

Let $A=\{1,2,3\}$ and let $R=\{(1,1),(2,2),(3,3),(1,2),(2,1),(2,3),(3,2)\} .$ Then, $R$ is

A. reflexive and symmetric but not transitive

B. symmetric and transitive but not reflexive

C. reflexive and transitive but not symmetric

D. an equivalence relation

 

Solution:

Given set $A=\{1,2,3\}$

And $R=\{(1,1),(2,2),(3,3),(1,2),(2,1),(2,3),(3,2)\}$

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, $(1,1) \in R,(2,2) \in R,(3,3) \in R$

Therefore, $R$ is reflexive $\ldots \ldots$ (1)

Check for symmetric

Since, $(1,2) \in R$ and $(2,1) \in R$

$(2,3) \in R$ and $(3,2) \in R$

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

Check for transitive

Here,$(1,2) \in R$ and $(2,3) \in R$ but $(1,3) \notin R$

Therefore, $\mathrm{R}$ is not transitive ....... (3)

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

Correct option will be (A)

 

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