De Morgan's Laws (Set Theory)/Relative Complement/Complement of Intersection

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Theorem

Let $S, T_1, T_2$ be sets such that $T_1, T_2$ are both subsets of $S$.


Then, using the notation of the relative complement:

$\complement_S \left({T_1 \cap T_2}\right) = \complement_S \left({T_1}\right) \cup \complement_S \left({T_2}\right)$


Proof

Let $T_1, T_2 \subseteq S$.

Then from Intersection is Subset and Subset Relation is Transitive:

$T_1 \cap T_2 \subseteq S$

Hence:

\(\displaystyle \complement_S \left({T_1 \cap T_2}\right)\) \(=\) \(\displaystyle S \setminus \left({T_1 \cap T_2}\right)\) Definition of Relative Complement
\(\displaystyle \) \(=\) \(\displaystyle \left({S \setminus T_1}\right) \cup \left({S \setminus T_2}\right)\) De Morgan's Laws: Difference with Intersection
\(\displaystyle \) \(=\) \(\displaystyle \complement_S \left({T_1}\right) \cup \complement_S \left({T_2}\right)\) Definition of Relative Complement

$\blacksquare$


Source of Name

This entry was named for Augustus De Morgan.


Sources