Set Union is not Cancellable

Theorem
Set union is not a cancellable operation.

That is, for a given $A, B, C \subseteq S$ for some $S$, it is not always the case that:


 * $A \cup B = A \cup C \implies B = C$

Proof
Proof by Counterexample:

Let $S = \left\{ {a, b}\right\}$.

Let:
 * $A = \left\{ {a, b}\right\}$
 * $B = \left\{ {a}\right\}$
 * $C = \left\{ {b}\right\}$

Then:

but: