logic problem regarding symmetric difference

Question

symmetric difference means: given the sets $A, B$, then: $$A \triangle B = (A\cup B)\setminus (A \cap B) = (A\setminus B) \cup (B\setminus A)$$

We'll say that $x\in (A\cap B) \Longrightarrow x\notin (A\triangle B) $.

Now, given 3 sets: $A, B, C$, I was being asked to prove that:

if $A\cap B \subseteq A\triangle B \triangle C \Longrightarrow A\cap B \subseteq C$.

at a different question I was asked to determine whether the following argument if True or False:

$$x \in A\triangle B \triangle C \to x\notin A\cap B$$

normally I would say this argument is true, but after I proved the previous argument, I'm not sure anymore. is that mean that the statement is False as there exists a contradiction, although it contradicts the definition of a symmetric difference?

Answer 1

normally I would say this argument is true, but after I proved the previous argument, I'm not sure anymore.

Yes, don't be sure.

Note: $$A\triangle B\triangle C=(A\cap B^\complement\cap C^\complement)\cup(A^\complement\cap B\cap C^\complement)\cup(A^\complement\cap B^\complement\cap C)\cup(A\cap B\cap C)$$

Answer 2

When you open the last formula, should you get:

1. $x$ is either an element of $C$ exclusive $\triangle $ OR
2. $x$ is an element of $ \triangle \setminus C$ which implies that $x \notin \cap $.

But for 1. if $ \cap \subseteq C$ then $x \in A\triangle B \triangle C $ and $x \in A\cap B$

Answer 3

Hint:

With e.g. induction it can be proved that $x\in A_1\Delta\cdots\Delta A_n$ if and only if the number of elements of set $\{i\in\{1,\dots,n\}\mid x\in A_i\}$ is odd.