De Morgan's Laws (Set Theory)/Set Complement

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Theorem

Let $T_1, T_2$ be subsets of a universe $\mathbb U$.

Let $\overline T_1$ denote the set complement of $T_1$.


Then:

Complement of Intersection

$\overline {T_1 \cap T_2} = \overline T_1 \cup \overline T_2$


Complement of Union

$\overline {T_1 \cup T_2} = \overline T_1 \cap \overline T_2$


General Case

Let $\mathbb T$ be a set of sets, all of which are subsets of a universe $\mathbb U$.


Then:

Complement of Intersection

$\displaystyle \complement \paren {\bigcap \mathbb T} = \bigcup_{H \mathop \in \mathbb T} \complement \paren H$


Complement of Union

$\displaystyle \complement \paren {\bigcup \mathbb T} = \bigcap_{H \mathop \in \mathbb T} \complement \paren H$


Family of Sets

Let $\left\langle{S_i}\right\rangle_{i \in I}$ be a family of sets, all of which are subsets of a universe $\mathbb U$.

Then:

Complement of Intersection

$\displaystyle \map \complement {\bigcap_{i \mathop \in I} S_i} = \bigcup_{i \mathop \in I} \map \complement {S_i}$


Complement of Union

$\displaystyle \map \complement {\bigcup_{i \mathop \in I} S_i} = \bigcap_{i \mathop \in I} \map \complement {S_i}$


Source of Name

This entry was named for Augustus De Morgan.