$\begin{array}{1 1} \text{Only Symmetric} \\ \text{Only Transitive} \\ \text{Only Reflexive} \\ \text{Symmetric and Transitive, not Reflexive} \end{array} $

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Show that the relation \(R\) in the set {1,2,3} given by R={(1,2), (2,1)} is symmetric but neither reflexive nor transitive.

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

Let $A$ bet a set with 3 elements {$1,2,3$}. Given that that the relation R in the set given by R={(1,2), (2,1)}:

Given $A=\{1,2,3\}$, we can see that $(1,1), (2,2)$ and $(3,3)$ are $\not \in R=\{(1,2)(2,1)\}$. Hence $R$ is not reflexive.

Given $(1,2)\in R\;and\;(2,1)\in R$ $R$ is symmetric.

We know that a relation R in a set A is called transitive. if $(a_1,a_2) \in\; R$ and $(a_2,a_3)\in R \Rightarrow \;(a_1,a_3)\in R\; $for all $\; a_1,a_2,a_3 \in A$

Here, assuming $a=1, b=2, c=1$, $ (a,b) \in R \rightarrow (1,2) \in R$ and $(b,c) \in R \rightarrow $ $(2,1) \in R$.

However, $(a,c)$, i.e., $(1,1) \not \in R$. Hence $R$ is not transitive.

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