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Give an example of a relation which is (iii) Reflexive and symmetric but not transitive.

- 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 $\mathbf{symmetric}$, if $(a_1,a_2) \in R\;\Rightarrow\; (a_2,a_1)\in R \; for \;a_1,a_2 \in A$

In a set of integers, we can observe that the Relation $R = \{(a,b): a \leq b$ does not satisify the properties of a symmetric relation.

For example, if $a = 1, b=2$, we can see that while $(1,2) \in R$ because $1 \leq 2$, $(2,1) \not \in R$ because $2 \not \leq 1$. Therefore, $R$ is not symmetric.

Given the Relation $R = \{(a,b): a \leq b$, we can observe that if $a=b, a \leq a$ which is true. Therefore, R is reflexive.

Given the Relation $R = \{(a,b): a \leq b$ we can further observe that if $a=1, b=2, c=3$, $(a,b) = (1,2) \in R$, and $(b,c) = (2,3)$ is in $R$ because $1 \leq 2$ and $2 \leq 3$ are true.Similarly $(a,c) = (1,3) \in R$ is true, because $1 \leq 3$. erefore $R$ is transitive..

Therefore, we can conclude that the Relation $R = \{(a,b): a \leq b$ is transitive and reflexive but not symmetrical.

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