Union of Empty Set
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Theorem
Consider the set of sets $\mathbb S$ such that $\mathbb S$ is the empty set $\O$.
Then the union of $\mathbb S$ is $\O$:
- $\mathbb S = \O \implies \bigcup \mathbb S = \O$
Proof
Let $\mathbb S = \O$.
Then from the definition:
- $\bigcup \mathbb S = \set {x: \exists X \in \mathbb S: x \in X}$
from which it follows directly:
- $\bigcup \mathbb S = \O$
as there are no sets in $\mathbb S$.
$\blacksquare$
Also presented as
Using the terminology of indexed families, this can also be written as:
- $\ds \bigcup_{i \mathop \in \O} S_i = \set {x: \exists i \in \O: x \in S_i} = \O$
Also see
Sources
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 4$: Unions and Intersections
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 9$: Families
- 1965: Claude Berge and A. Ghouila-Houri: Programming, Games and Transportation Networks ... (previous) ... (next): $1$. Preliminary ideas; sets, vector spaces: $1.1$. Sets
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 3$: Unions and Intersections of Sets: Exercise $3.7$
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Unions and Intersections
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 4$: Indexed Families of Sets
- 1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.4$: Sets of Sets: Exercise $1.4.2: \ \text{(ii)}$
- 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 1$: Fundamental Concepts
- 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): Appendix $\text{A}$: Set Theory: Families