Answer
\[
\boxed{\text{True. If } A \triangle C = B \triangle C,\ \text{then } A = B}
\]
Work Step by Step
We are given:
> If \( A \triangle C = B \triangle C \), then \( A = B \)
Let’s prove this statement is **true** using properties of symmetric difference.
---
### ✅ Useful fact:
The symmetric difference satisfies the identity:
\[
A \triangle C = B \triangle C \iff A = B
\]
This is **true** because symmetric difference is **cancellable**, similar to addition:
\[
A \triangle C = B \triangle C \iff A \triangle (C \triangle C) = B
\Rightarrow A = B
\]
Since:
\[
C \triangle C = \emptyset \quad \text{(from problem 50)}
\Rightarrow A \triangle \emptyset = A \quad \text{(from problem 48)}
\]
So:
\[
A \triangle C = B \triangle C
\Rightarrow A = B
\]
