# 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.

This question has appeared in model paper 2012

Toolbox:
• 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.
answered Feb 21, 2013 by
edited Mar 8, 2013