If $A$, $B$, and $C$ are sets, the only way that $A\cup C = B \cup C$ is if $A=B$

Question

If $A$, $B$, and $C$ are three sets, then the only way that $A\cup C$ can equal $B\cup C$ is $A = B$.

I believe this statement is false and here is why:

Let $A=\{1\}$, $B=\{2\}$, and $C=\{1,2,3,4\}$. In this scenario $A\cup C=\{1,2,3,4\}$ and $B\cup C=\{1,2,3,4\}$ however, $A\ne B$. Making the statement false.

Have I explained this correctly?

Answer 1

Your example is the proof that the statement is false.