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

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:


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


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

General Case
Let $T$ be a subset of $S$.

Let $\mathcal P \left({T}\right)$ be the power set of $T$.

Let $\mathbb T \subseteq \mathcal P \left({T}\right)$.

Then:


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


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

Proof
Let $T_1, T_2 \subseteq S$.

Then:

So we can talk about $\complement_S \left({T_1 \cap T_2}\right)$ and $\complement_S \left({T_1 \cup T_2}\right)$.

Hence the following results are defined: