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Give an example of a relation which is (ii) Transitive but neither reflexive nor symmetric.

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- A relation R in a set A is called $\mathbf{ reflexive},$ if $(a,a) \in R\;$ for every $\; a\in\;A$
- A relation R in a set A is called $\mathbf{symmetric}$, if $(a_1,a_2) \in R\;\Rightarrow\; (a_2,a_1)\in R \; for \;a_1,a_2 \in A$
- A relation R in a set A is called $\mathbf{transitive},$ if $(a_1,a_2) \in R$ and $(a_2,a_3) \in R \; \Rightarrow \;(a_1,a_3)\in R$ for all$\; a_1,a_2,a_3 \in A$

A relation R in a set A is called $\mathbf{transitive},$ if $(a_1,a_2) \in R$ and $(a_2,a_3) \in R \; \Rightarrow \;(a_1,a_3)\in R$ for all$\; a_1,a_2,a_3 \in A$

In a set of integers, we can observe that the Relation $R = \{(1,1), (2,2) (3,3\}$ does not satisify the properties of a transitive relation.

Given that the Relation $R = \{(1,1), (2,2) (3,3)\}$, we can observe that if $a=1,2$ or $3$, $(1,1), (2,2), (3,3) \in R$. Therefore, R is reflexive.

Given that the Relation $R = \{(1,1), (2,2) (3,3)\}$, we can further observe that $(a,b) = (1,1) \in R$, and $(b,a) = (1,1)$ is in $R$. Similarly $(2,2) \in R$ and $(3,3) \in R$. Therefore $R$ is symmetric.

Therefore, we can conclude that the Relation $R = \{(1,1), (2,2) (3,3\}$ is not transitive but reflexive and symmetrical.

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