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

(NONE) Given relation $R = \{(x, y) : x \text{ is the wife of } y\}$.
$1$. Reflexivity:
For any human being $x \in A$,$x$ cannot be the wife of herself. Thus,$(x, x) \notin R$ for any $x \in A$.
Therefore,$R$ is not reflexive.
$2$. Symmetry:
Let $(x, y) \in R$. This implies $x$ is the wife of $y$. This means $y$ must be the husband of $x$. Since $y$ is a husband,$y$ cannot be the wife of $x$. Thus,$(y, x) \notin R$.
Therefore,$R$ is not symmetric.
$3$. Transitivity:
Let $(x, y) \in R$ and $(y, z) \in R$. This implies $x$ is the wife of $y$ and $y$ is the wife of $z$. Since $y$ is the wife of $z$,$y$ must be a female. However,if $x$ is the wife of $y$,$y$ must be a male. This is a contradiction. Thus,the condition $(x, y) \in R$ and $(y, z) \in R$ can never be satisfied simultaneously. Therefore,$R$ is not transitive.
Conclusion: $R$ is neither reflexive,nor symmetric,nor transitive.