explain transitive nature of the given set

Question

A relation $R$ in the set of human beings in a town given by

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

How is it transitive? Can you explain?

Answer 1

$R$ is transitive if the following is true: whenever $\langle x,y\rangle\in R$ and $\langle y,z\rangle\in R$, then $\langle x,z\rangle\in R$. In other words, if $x$ is the wife of $y$, and $y$ is the wife of $z$, then $x$ is the wife of $z$.

If we assume that all married couples in that town consist of a husband and a wife, then $R$ is vacuously transitive: it never happens that $x$ is the wife of $y$ and $y$ is the wife of $z$, so the condition of transitivity doesn’t actually impose any restriction on $R$. To put it a little differently, in this town you simply can’t find a violation of transitivity: there are no three people $x,y$, and $z$ such that $x$ is the wife of $y$, $y$ is the wife of $z$, and $x$ is not the wife of $z$ for the simple reason that there are no three people $x,y$, and $z$ such that $x$ is the wife of $y$ and $y$ is the wife of $z$ in the first place.

If, however, the town has a pair of women, say $x$ and $y$, who are each other’s wives, then $R$ is not transitive. To see this, note that $\langle x,y\rangle\in R$, and $\langle y,x\rangle\in R$, so transitivity would require that $\langle x,x\rangle\in R$. But $x$ is not her own wife, so this is not the case.

Transitivity can also fail in a society that allows marriages of more than two people.