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

$\blacksquare$


Sources