Set Difference Relative Complement

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Theorem

Let $A, B \subseteq S$.

Then the set difference between $A$ and $B$ can be expressed as the intersection with the relative complement with respect to $S$:

$A \setminus B = A \cap \complement_S \left({B}\right)$

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle A \setminus B$	$=$	$\displaystyle \left\{ {x \in A: x \notin B}\right\}$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of set difference
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \left\{ {x \in A: x \in \complement_S \left({B}\right)}\right\}$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of relative complement
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle A \cap \complement_S \left({B}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of set intersection

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle A \setminus B\)	\(=\)	\(\displaystyle \left\{ {x \in A: x \notin B}\right\}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of set difference
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left\{ {x \in A: x \in \complement_S \left({B}\right)}\right\}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of relative complement
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle A \cap \complement_S \left({B}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of set intersection