If $R=\{(x,y): x\text{ is wife of } y\}$, then is $R$ transitive? If $R=\{(x,y): x\text{ is wife of } y\}$, determine whether the relation $R$ is transitive or not.


My Try: For Transitivity, If $(a,b) \in R$ and $(b,c)\in R\;,$ Then $(a,c)\in R.$.

Here If $x$ is a wife of $y$, then $y$ is not a wife of $z$. Therefore $x$ is not a wife of $z$.

Means $(x,y)\in R$ and $(y,z)\notin R$, then $(x,z)\notin R$

So $R$ is not a Transitive Relation.

But in Book, It is Transitive Relation.

Plz explain me with example. How this can be possible? Thanks
The best explanation is probably that the possibility of same-sex marriages were not on the author's mind when he/she came up with the exercise.

If we suppose that marriages is always between a man and a woman (and genders are binary, bla bla bla), then the relation is indeed transitive, but vacuously so.

The condition it has to satisfy is that if $x$ is the wife of $y$ and $y$ is the wife of $z$, then $x$ is the wife of $z$. However this is true because the premise (i.e. the part after "if") is impossible to satisfy (under our assumed assumptions): $x$ can only be the wife of $y$ if $y$ is a man, and $y$ can only be the wife of $z$ if $y$ is a woman. Since nothing is both a man and a woman, both of these cannot be true at the same time.

In other words, the only way for a relation not to be transitive is if there is some $x$, $y$, and $z$ such that $xRy$ and $yRz$ and not $xRz$. If there is no such example, the relation is transitive -- no matter why there is no example.

Thus, the relation is transitive for the same reason that the empty relation is transitive: Because there are no triples $(x,y,z)$ such that $xRy$ and $yRz$, the condition for being transitive is not automatically met.

On the other hand, if $R$ includes information about at least one lesbian marriage, then it fails to be transitive: Suppose Alice and Betty are married. Then, in the case $(x,y,z)=(\mathit{Alice},\mathit{Betty},\mathit{Alice})$ we have that $\mathit{Alice}\mathrel{R}\mathit{Betty}$ and $\mathit{Betty}\mathrel{R}\mathit{Alice}$, yet not $\mathit{Alice}\mathrel{R}\mathit{Alice}$, since Alice is not her own wife.
It is transitive, considering that a set (y,z) (implying that y who is wife of x, is husband of z) does not exist.
To prove transitive condition you need to have the set (x,y) and also (y,z).Then you find out whether (y,z) is in the relation to figure out the transitive condition.
And if the condition (y,z) does not exist (like in the above case) you don't need test for transitiveness. It is trasitive by default.