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?
Best Answer
Yes, don't be sure.
Note: $$A\triangle B\triangle C=(A\cap B^\complement\cap C^\complement)\cup(A^\complement\cap B\cap C^\complement)\cup(A^\complement\cap B^\complement\cap C)\cup(A\cap B\cap C)$$