I have to prove or disprove the following claim:
If $A$, $B$ and $C$ are sets, where $A \triangle C = B \triangle C $, then $ A = B $
My proof is the following:
Consider any sets $A$, $B$ and $C$, where $A \triangle C = B \triangle C $. We need to prove, that $ A = B $ by showing that $ A \subseteq B $ and $ B \subseteq A $.
First, we will show that $ A \subseteq B$. For this, consider any $ x \in A $. We will prove that $ x \in B $. Notice, that either $ x \in C $ or $ x \not\in C $. Therefore we will proceed by cases:
- $ x \in C $. Since $ x \in A $ and $ x \in C$, we see that $ x \not\in A \triangle C $. Consequently, as $ A \triangle C = B \triangle C $, we know that $ x\not\in B \triangle C $. This means that $ x \in B $.
- $ x \not\in C $. This means that $ x \in A \triangle C $. Consequently, $ x \in B \triangle C $. Therefore $ x \in B $.
The second part of the proof (to show that $ B \subseteq A $) is very similar to this.
Can someone please take a look at the proof and criticize it? I'm especially interested in this part:
Note, that either $ x \in C $ or $ x \not\in C $. Therefore we will proceed by cases:
Is it okay to proceed this way or is it unclear or incorrect? I'd be also grateful if someone could provide another simpler proof if there exists one.