Intersection with Relative Complement is Empty

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Theorem

The intersection of a set and its relative complement is the empty set:

$T \cap \relcomp S T = \O$


Proof

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

$\blacksquare$


Sources