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

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

Then:


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


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

where $\overline T_1$ is the set complement of $T_1$.

It is arguable that this notation may be easier to follow:


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


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


 * DeMorganComplementIntersection.png DeMorganComplementUnion.png

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

Then:
 * $(1): \quad \displaystyle \complement \left({\bigcap \mathbb T}\right) = \bigcup_{H \in \mathbb T} \complement \left({H}\right)$


 * $(2): \quad \displaystyle \complement \left({\bigcup \mathbb T}\right) = \bigcap_{H \in \mathbb T} \complement \left({H}\right)$