Let R be a relation on a set X and if we have to prove that it is not a transitive relation,

then we have to show that there exists a,b,c ∈ X such that

 (a,b), (b,c) ∈ R ⇏ (a,c) ∈ R 

Now given relation R is 

R = {(x,y) : x is the wife of y}

Now there is no z such that (y,z) ∈ R.

Since we are unable to show that  (x,y), (y,z) ∈ R ⇏ (x,z) ∈ R

Therefore R is a transitive relation.