Set Difference with Intersection is Difference

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Theorem

The set difference with the intersection is just the set difference.

Let $S, T$ be sets.

Then:

$S \setminus \paren {S \cap T} = S \setminus T$

Proof

 $\displaystyle S \setminus \paren {S \cap T}$ $=$ $\displaystyle \paren {S \setminus S} \cup \paren {S \setminus T}$ De Morgan's Laws: Difference with Intersection $\displaystyle$ $=$ $\displaystyle \O \cup \paren {S \setminus T}$ Set Difference with Self is Empty Set $\displaystyle$ $=$ $\displaystyle S \setminus T$ Union with Empty Set

$\blacksquare$