Set Difference Intersection with Second Set is Empty Set

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Theorem

Let $S$ and $T$ be sets.

The intersection of the set difference of $S$ and $T$ with $T$ is the empty set:

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


Proof 1

\(\displaystyle \paren {S \setminus T} \cap T\) \(=\) \(\displaystyle \paren {S \cap T} \setminus \paren {T \cap T}\) Set Intersection Distributes over Set Difference
\(\displaystyle \) \(=\) \(\displaystyle \paren {S \cap T} \setminus T\) Intersection is Idempotent
\(\displaystyle \) \(=\) \(\displaystyle \O\) Set Difference of Intersection with Set is Empty Set

$\blacksquare$


Proof 2

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

$\blacksquare$


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