De Morgan's Laws (Set Theory)

Theorem
De Morgan's laws, or the De Morgan formulas, etc. are a collection of results in set theory as follows.

Relative Complement
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:


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


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

Set Complement
When $T_1, T_2$ are understood to belong to a universe $\mathbb U$, the notation of the set complement can be used:


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


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

It is arguable that this notation is 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 Result
Let $T$ is a subset of $S$.

Then:


 * $\displaystyle \complement_S \left({\bigcap \mathbb T}\right) = \bigcup_{T' \in \mathbb T} \complement_S \left({T'}\right)$


 * $\displaystyle \complement_S \left({\bigcup \mathbb T}\right) = \bigcap_{T' \in \mathbb T} \complement_S \left({T'}\right)$

In the context of set complement:


 * $\displaystyle \complement \left({\bigcap \mathbb T}\right) = \bigcup_{T' \in \mathbb T} \complement \left({T'}\right)$


 * $\displaystyle \complement \left({\bigcup \mathbb T}\right) = \bigcap_{T' \in \mathbb T} \complement \left({T'}\right)$

Relative Complement
Let $T_1, T_2 \subseteq S$.

Then:
 * $T_1 \cap T_2 \subseteq S$ from Intersection Subset and Subsets Transitive;
 * $T_1 \cup T_2 \subseteq S$ from Union Smallest.

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:

General Proof
It is necessary only to prove:


 * $\displaystyle S \setminus \bigcap \mathbb T = \bigcup_{T' \in \mathbb T} \left({S \setminus T'}\right)$


 * $\displaystyle S \setminus \bigcup \mathbb T = \bigcap_{T' \in \mathbb T} \left({S \setminus T'}\right)$

... as the others follow directly from the definition of relative complement and set complement.

First result
To prove that:
 * $\displaystyle S \setminus \bigcap \mathbb T = \bigcup_{T' \in \mathbb T} \left({S \setminus T'}\right)$:

Suppose:
 * $\displaystyle x \in S \setminus \bigcap \mathbb T$

Note that by Set Difference Subset we have that $x \in S$ (we need this later).

Then:

Therefore:
 * $S \setminus \bigcap \mathbb T = \bigcup_{T' \in \mathbb T} \left({S \setminus T'}\right)$

Second result
To prove that:
 * $S \setminus \bigcup \mathbb T = \bigcap_{T' \in \mathbb T} \left({S \setminus T'}\right)$

Suppose:
 * $\displaystyle x \in S \setminus \bigcup \mathbb T$

Note that by Set Difference Subset we have that $x \in S$ (we need this later).

Then:

Therefore:
 * $\displaystyle S \setminus \mathbb T = \bigcap_{T' \in \mathbb T} \left({S \setminus T'}\right)$

Strictly speaking, these are not the actual laws he devised, but an application of those laws in the context of set theory.

This result is known by some authors, for example, as the duality principle.