Set Intersection Not Cancellable

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Theorem

Let $S$ be a set and let $\mathcal P \left({S}\right)$ be the power set of $S$.

Let $S_1, S_2, T \in \mathcal P \left({S}\right)$.

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 $S = \left\{{1, 2, 3}\right\}$.

Let $T = \left\{{3}\right\}$.

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

Then $S_1 \cap T = S_2 \cap T = \left\{{3}\right\}$ but $S_1 \ne S_2$.

$\blacksquare$