TITLE: Let A,B,C be sets. If A△B=A△C, does this imply that B=C?
QUESTION [6 upvotes]: I believe that a counterexample exists, but I have struggled to find one since I need to satisfy (A\B) U (B\A) = (A\C) U (C\A), which is a very complex statement. Also I used △ to represent symmetric difference.

REPLY [2 votes]: I'd like to add one more proof -- by contradiction. Suppose there exist such $x \in B$ that $x \notin C$. Consider two cases
$$
x \in A 
\xrightarrow{x \notin C} x \in A \triangle C
\xrightarrow{A \triangle C = A \triangle B} x \in A \triangle B
\xrightarrow{x \in A} x \notin B
~~~~~~\text{(contradiction)}
$$
$$
x \notin A 
\xrightarrow{x \in B} x \in A \triangle B
\xrightarrow{A \triangle C = A \triangle B} x \in A \triangle C
\xrightarrow{x \notin A} x \in C
~~~~~~\text{(contradiction)}
$$
Swapping letters 'B' and 'C' we can prove that there is also no such $x$ that $x \notin B$ and $x \in C$.
This completes the proof.