Make the set $R$ transitive

Question

Let $R$ be a relation on a set $A=\{w,x,y,z\}$ defined by $R=\{(w,x),(y,x),(x,y),(z,z)\}$. Using the original relation, $R$, make the necessary minimal additions to make $R$ transitive.

I thought that I would only have to add the element $(x,x)$ to $R$ to make a transitive set. My logic is as follows:

1. $w \to x \to x$
2. $y \to x \to x$
3. $x \to y \to x \to x$
4. $z \to z$

But this answer is not correct. The given solution involves adding the elements $(w,y),(y,y),(x,x)$.

Can someone please explain why this is the case? Thank you!

Answer 1

Since $w$ is related to $x$ And $x$ is related to $y$ Therefore for transitivity you need $w$ To be related to $y$ And so on....