I have a couple of statements to prove (self-learning, not homework). I'm not able to proceed with any of them (I've tried starting from RHS, LHS, etc.). I suppose I'm missing something. I would like to receive some tips or a solution for one of them and I will try to solve the rest.
The symmetric difference is defined as follows: $A \triangle B = (A \setminus B) \cup (B \setminus A)$.
$(A \cup B) \triangle C = (A \triangle C) \triangle (B \setminus A)$.
$(A \cap B) \triangle C = (A \triangle C) \triangle (A \setminus B)$.
$(A \setminus B) \triangle C = (A \triangle C) \triangle (A \cap B)$.
Best Answer
E.g. the first one; try to show two inclusions.
Suppose $x \in (A\cup B)\Delta C$. Then there are two cases:
So the left to right inclusion has been shown now.
The right to left inclusion has similar cases.
Try it, and then the rest. Verify the claims first by drawing a Venn diagram for three sets in general position.