Intersection with Relative Complement

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Theorem

$T \cap \complement_S \left({T}\right) = \varnothing$

	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle T \cap \complement_S \left({T}\right)$	$=$	$\displaystyle \left({S \setminus T}\right) \cap T$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of relative complement
	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \varnothing$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Set Difference Intersection Second Set is Empty Set

$\blacksquare$

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle T \cap \complement_S \left({T}\right)\)	\(=\)	\(\displaystyle \left({S \setminus T}\right) \cap T\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of relative complement
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \varnothing\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Set Difference Intersection Second Set is Empty Set