I suppose that if I can prove that every element that belongs to set $A$ also belongs to set $B$ and vice versa and also any element that doesn't belong to set $A$ doesn't belong to set $B$ either and vice versa, then I can prove that $A = B$.
From Venn diagrams, it is clear that an element that belongs to ($A \Delta C$) then it is either in set $A$ or in set $C$ but not in both. Similarly, if the element is not in ($A \Delta C$), then either it doesn't belong to either or belongs to the common region in the Venn diagram. But I'm unable to see how to proceed from here.
$X \Delta Y = (X-Y) \cup (Y-X)$
Also this problem seems trivial to me. But formally proving seems difficult.
Best Answer
Suppose $x \in A$. Then $x \in A \setminus C$ or $x \in A \cap C$.
Case 1. $x \in A \setminus C$. Then, since $A \Delta C = B \Delta C = (B\setminus C) \cup (C \setminus B)$, $x \in B \setminus C$ as $x \not\in C$. In particular, $x \in B$.
Case 2. $x \in A \cap C$. Again, since $A \Delta C = B \Delta C$, $x \not\in B \Delta C$. That is, $x \in B \cap C$ or $x \not\in B \cup C$. Since $x \in C$, it must be the case that $x \in B \cap C$. In particular, $x \in B$.
Hence, $A \subseteq B$. The other direction is similar.