Question on set theory and first order logic

Question

I need help with this problem on set theory.

For any sets $A$ and $B$, consider the set $S$ defined below:

$$S = \{ x \mid \neg (x ∈ A \to x ∈ B) \}$$

I need to write an expression for $S$ in terms of $A$ and $B$ using the standard set operators (union, intersection, etc.)

Answer 1

Hint $$p\to q \iff \lnot p\lor q$$ and $$\{x\mid x\in A \land x\in B\}=A\cap B$$ $$\{x\mid x\in A \lor x\in B\}=A\cup B$$ $$\{x\mid x\in \lnot(x\in A)\}=A^C$$ where $A^C$ is complement of $A$.