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Give an example of a relation which is (iv) Reflexive and transitive but not 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$\; a_1,a_2,a_3 \in A$

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

For example, if $a = b = 1$, we can see that $(-1,-1) \not \in R$. Hence it is not reflexive.

Given the Relation $R = \{(-1,-2), (-2, -3), (-2,-1), (-3,1), (-1,-3), (-3,-2)\}$, we can see that for any of the ordered pairs $(a,b) \in R \rightarrow (b,a) \in R$. Therefore R is symmetric.

Given the Relation $R = \{(-1,-2), (-2, -3), (-2,-1), (-3,1), (-1,-3), (-3,-2)\}$, we can observe that for any two pair of ordered pairs $(a,b), (b,c) \in R \rightarrow (a,c) \in R$.

Therefore, we can conclude that the Relation s transitive and symmetrical but not reflexive.

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