I'm studying for a test and there's a question that I've tried and I don't understand:
Let $E$ be a binary relation on a set $A$; let a binary relation $F$ on $\mathcal P (A) \setminus \{\emptyset\}$ be: $(B,C) \in F \iff \exists b \in B, \exists c \in C : (b,c)\in E$. If $E$ is transitive is $F$ transitive?
(I need to prove it if so).
The relation E={{1,2},{3,4}} already disproves the first point, it seems, since {1} and {2,3} are related by F and {2,3} and {4} are related by F but {1} and {4} are not.