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 $S = \set {a, b}$.

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

Then:

but: