# Determine whether each of the following relations are reflexive, symmetric and transitive: Relation $R$ in the set $A$ of all human beings in a town at a particular time given by $R=\{(x,y):\; \text{x is the wife of y}\}$

Toolbox:
• A relation R in a set A is called reflexive. if $(a,a) \in R\;$ for every$\;a \in A$
• A relation R in a set A is called symmetric. if $(a_1,a_2) \in R\;=>\; (a_2,a_1)\in R \; for \;a_1,a_2 \in A$
• A relation R in a set A is called transitive. if $(a_1,a_2) \in\; R$ and $(a_2,a_3)\in R=>\;(a_1,a_3)\in R for\; all\; a_1,a_2,a_3 \in A$
Given the set $A$ of all human beings in a town at a particular time and $R=\{(x,y):\; \text{x is the wife of y}\}$ :
Since $x$ is not the wife of $x, \; (x,x) \not \in R.$ Hence $R$ is not reflexive.
If $x$ is the wife of $y$ and $y$ cannot be the wife of $x$, i.e, $(x,y) \in R$ but $(y,x) \not \in R$. Hence $R$ is not symmetric.
If $x$ is the wife of $y$ then $y$ cannot be the wife of anybody because he is male and only be a husband. For $R$ to be  transitive, if x is related with y and y is not related with any other element then it is transitive. Hence $R$ is  transitive.

edited Mar 8, 2013