Set Intersection Not Cancellable

Theorem

Let $S$ be a set and let $\powerset S$ be the power set of $S$.

Let $S_1, S_2, T \in \powerset S$.

Suppose that $S_1 \cap T = S_2 \cap T$.

Then it is not necessarily the case that $S_1 = S_2$.

Proof

Proof by Counterexample:

Let $S = \set {1, 2, 3}$.

Let $T = \set 3$.

Let $S_1 = \set {1, 3}, S_2 = \set {2, 3}$

Then $S_1 \cap T = S_2 \cap T = \set 3$ but $S_1 \ne S_2$.

$\blacksquare$