Set Difference with Set Difference

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Theorem

The set difference with the set difference of two sets is the intersection of the two sets:

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


Proof

\(\displaystyle S \setminus \paren {S \setminus T}\) \(=\) \(\displaystyle \paren {S \setminus S} \cup \paren {S \cap T}\) Set Difference with Set Difference is Union of Set Difference with Intersection
\(\displaystyle \) \(=\) \(\displaystyle \O \cup \paren {S \cap T}\) Set Difference with Self is Empty Set
\(\displaystyle \) \(=\) \(\displaystyle S \cap T\) Union with Empty Set


Interchanging $S$ and $T$:

\(\displaystyle T \setminus \paren {T \setminus S}\) \(=\) \(\displaystyle T \cap S\)
\(\displaystyle \) \(=\) \(\displaystyle S \cap T\) Intersection is Commutative

$\blacksquare$


Sources