I'm given the set $ U=\{1,2,3,4 \}$ and then asked to determine $S\subset \mathcal P(U)\times \mathcal P(U)$, where $\mathcal P(U)$ is the power set of $U$ and $S$ is the solution set for the following equation system:
$\{ 1,3,4\} \cap A \triangle \{ 1,2 \} \cap B = \{ 1,2,3 \}$
$\{ 2,3,4\} \cap A \triangle \{ 1,3\} \cap B = \{ 2,3,4 \}$
I understand that the set $\mathcal P(U)$ along with the set operations $\triangle$ and $\setminus$ constitute a ring, therefore some arithmetic can be done using the properties of those operations, yet I do not know how to determine $S$. Any suggestion on how to tackle this kind of equations?
Taking the suggestion:
First line implies that $ (1\in A "XOR" 1\in B) $, indeed, by the definition of symmetric difference, if both $A$ and $B$ had 1 as an element, then it would not be an element of $ \{ 1,2,3 \} $, also, if 1 was not an element of $A$ nor $B$ then again it would not be an element of $ \{ 1,2,3 \} $. Using the prior reasoning for each element we get the following:
$(1\in A "XOR" 1\in B)\wedge (2\in B) \wedge (3\in A) \wedge (4 \notin A)$
and
$(1\notin B)\wedge (2\in A) \wedge (3\in A "XOR"3\in B) \wedge (4 \in A)$
Then notice we have $(4 \in A) \wedge (4 \notin A)$, therefore $$ S=\emptyset $$