How to prove $(a \setminus b) ∪ c= (a ∪ c) \setminus (b ∪ c)$?

Question

How to prove $(a \setminus b) ∪ c= (a ∪ c) \setminus (b ∪ c)$? I tried to do some properties, but I don't know what to do next. 1

Answer 1

We show the equality of two sets $S_1$ and $S_2$ by showing that $S_1 \subseteq S_2$ and that $S_2 \subseteq S_1$.

So take an arbitrary element from the left-hand side and show that it is also an element of the set on the right-hand side. And after that, take an arbitrary element from the right-hand side and show that it is also an element of the set on the left-hand side.

Answer 2

Interesting that you tried to make sens of the equation as a whole. I would start from one end and try to come out from the other : $ (A \setminus B) \cup C = \{x, (x \in A \wedge x\not\in B) \vee x \in C\}$ $=\{x,(x\in A \vee x\in C)\wedge (x\not\in B \vee x \in C)\}=(A\cup C) \cap (\overline{B} \cup C),$ whereas you can esaily see that $(A \cup C) ∖ (B\cup C)=(A \cup C) \cap (\overline{B \cup C})$. You can clearly see here that the two sets are not the same. Any counter examples ?