Given a set $A$, define relation $R$ on $\mathcal{P}(A)$ by $\{(U,V) \in \mathcal{P}(A) \times \mathcal{P}(A) \colon \dots \}$. Is $R$ transitive? Given a set $A$, define relation $R$ on $\mathcal{P}(A)$  by
$$R = \{(U,V) \in \mathcal{P}(A) \times \mathcal{P}(A) \colon (U \cap V \neq \emptyset) \lor (U \cup V = \emptyset)\}.$$
I want to check if $R$ is a transitive relation.
My scratch work so far:
Let $A = \{a, b, c, d, e\}$ then $\mathcal{P}(A)=\{\{\emptyset\}, \{a\},\{b\},\{c\},\{d\},\{e\},\dots,\{a,b,c,d,e\}\}$.
So I know I want to prove that $URV \land VRW \Longrightarrow VRW$ for transitivity.
If I let $(U,V)=(\{a\},\{a,b\})$ then $(U \cap V) = \{a\} \neq \emptyset$, so that  $URV$,
$(V,W)=(\{a,b\},\{b\})$ then $(V \cap W) = \{b\} \neq \emptyset$, so that  $VRW$
Then $(U \cap V)\cap(V \cap W) = \emptyset$ right? Which means $URV∧VRW⟹V\not RW$.  Am I missing a key concept? Is this enough to prove it's not transitive? I know there exists subsets I can pull to show transitivity but if I show one isn't transitive, it should show the overall relation isn't transitive right?
Any feedback on this is appreciated. Thanks.
 A: Your argument is fine, I’m just writing the solution because you seem to have some little mistakes in your writing (I will pointed out)
To show that a relation $R$ is transitive, we need to show that
\begin{align*}
\forall x,y,z \, ,\, xRy \wedge yRz \implies xRz. \tag{1}
\end{align*}
In this case, if we let $A = \{a,b\}$, $U = \{a\}$, $V = \{a,b\}$ and $W = \{b\}$, then we have that
\begin{align*}
U \cap V = \{a\} \neq \emptyset \implies URV \\
V \cap W = \{b\} \neq \emptyset \implies VRW
\end{align*}
Although, we have that
\begin{align*}
U \cap W = \emptyset \implies U \not R W \tag{2}
\end{align*}
Therefore, you have that $R$ is not a transitive relation. $\square$

At $(1)$, when in your question you write the definition of a transitive relation, you say that for all $U,V,W \subseteq A$ we have that $URV \wedge VRW \implies VRW$ which is wrong, it should be $URV \wedge VRW \implies URW$.
At $(2)$, you check the intersection $(U \cap V) \cap (V \cap W)$ which is wrong. You only need to check the intersection between $U$ and $W$ (So, the intersection $U \cap W$) in order to say something about $URW$.

Finally, I just want to say that your thought is right, your arguments is correct. In order to show that $R$ is not transitive you only need to work out a counter example, as you well did. Continuation of a good work.
