Question

Let $A$ be the set of all human beings in town in a particular time. Then check whether the relation is reflexive, symmetric, transitive.

$A=\left\{\right(x,y):x\text{is wife of}y\}$.

Answer 1

Given the relation $A=\left\{\right(x,y):x\text{is wife of}y\}$.

This relation is not reflexive as $(x,x)\notin A$ since $x$ can't be its own wife.

Similarly the relation is not symmetric as if $(x,y)\in A$ this does not give $(y,x)\in A$ as if $x$ is wife of $y$ this does not imply $y$ is not wife of $x$.

Again the relation can't be transitive also.

As we can't let $(x,y)\in A$ and $(y,z)\in A$ since if $x$ is wife of $y$ then $y$ can not be wife of $z$.