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

Q.10 Give an example of a relation.

(ii) Which is transitive but neither reflexive nor symmetric.

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Let

$R = \left \{ \left ( x,y \right ): x> y \right \}$

Now for $x\in R$ ,$(x,x)\notin R$ so it is not reflexive.

Let $(x,y) \in R$  i.e. $x> y$

Then $y> x$ is not possible i.e. $(y,x) \notin R$ . So it is not symmetric.

Let $(x,y) \in R$  i.e. $x> y$    and  $(y,z) \in R$ i.e.$y> z$

we can write this as $x> y> z$

Hence,$x> z$  i.e. $(x,z)\in R$. So it is transitive.

Hence, it is transitive but neither reflexive nor symmetric.

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