Give an example of a relation. Which is Transitive but neither reflexive nor symmetric.

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1.Relation and Function

easy

Give an example of a relation. Which is Transitive but neither reflexive nor symmetric.

Option A

Option B

Option C

Option D

Solution

Consider a relation $R$ in $R$ defined as:

$R =\{( a , b ): a < b \}$

For any $a \in R$, we have $(a, a) \notin R$ since a cannot be strictly less than a itself.

In fact, $a=a$

$\therefore R$ is not reflexive.

Now, $(1,2)\in R$ $($ as $1<2)$

But, $2$ is not less than $1.$

$\therefore (2,1) \notin R$

$\therefore R$ is not symmetric.

Now, let $(a, b),\,(b, c) \in R$

$\Rightarrow a < b$ and $b < c$

$\Rightarrow a < c$

$\Rightarrow(a, c) \in R$

$\therefore R$ is transitive.

Hence, relation $R$ is transitive but not reflexive and symmetric.

Standard 12

Mathematics

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