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

Theorem

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

Then:

$\ds \map \complement {\bigcap \mathbb T} = \bigcup_{H \mathop \in \mathbb T} \map \complement H$

Proof

\(\ds \map \complement {\bigcap \mathbb T}\)

\(=\)

\(\ds \mathbb U \setminus \paren {\bigcap \mathbb T}\)

Definition of Set Complement

\(\ds \)

\(=\)

\(\ds \bigcup_{H \mathop \in \mathbb T} \paren {\mathbb U \setminus H}\)

De Morgan's Laws: Difference with Intersection

\(\ds \)

\(=\)

\(\ds \bigcup_{H \mathop \in \mathbb T} \map \complement H\)

Definition of Set Complement

$\blacksquare$

Source of Name

This entry was named for Augustus De Morgan.

Sources

• 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets
• 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.8$: Collections of Sets: Exercise $5$
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 7.1 \ \text{(i)}$: Unions and Intersections