Definition:Set Union/Set of Sets
Definition
Let $\mathbb S$ be a set of sets.
The union of $\mathbb S$ is:
- $\bigcup \mathbb S := \set {x: \exists X \in \mathbb S: x \in X}$
That is, the set of all elements of all elements of $\mathbb S$.
Thus the general union of two sets can be defined as:
- $\bigcup \set {S, T} = S \cup T$
Also denoted as
The symbol $\bigcup$ is rendered in display mode as $\ds \bigcup$.
Some sources denote $\bigcup \mathbb S$ as $\ds \bigcup_{S \mathop \in \mathbb S} S$.
Examples
Set of Arbitrary Sets
Let:
\(\ds A\) | \(=\) | \(\ds \set {1, 2, 3, 4}\) | ||||||||||||
\(\ds B\) | \(=\) | \(\ds \set {a, 3, 4}\) | ||||||||||||
\(\ds C\) | \(=\) | \(\ds \set {2, a}\) |
Let $\mathscr S = \set {A, B, C}$.
Then:
- $\bigcup \mathscr S = \set {1, 2, 3, 4, a}$
Set of Initial Segments
Let $\Z$ denote the set of integers.
Let $\map \Z n$ denote the initial segment of $\Z_{>0}$:
- $\map \Z n = \set {1, 2, \ldots, n}$
Let $\mathscr S := \set {\map \Z n: n \in \Z_{>0} }$
That is, $\mathscr S$ is the set of all initial segments of $\Z_{>0}$.
Then:
- $\bigcup \mathscr S = \Z_{>0}$
that is, the set of strictly positive integers.
Set of Unbounded Above Open Real Intervals
Let $\R$ denote the set of real numbers.
For a given $a \in \R$, let $S_a$ denote the (real) interval:
- $S_a = \openint a \to = \set {x \in \R: x > a}$
Let $\SS$ denote the family of sets indexed by $\R$:
- $\SS := \family {S_a}_{a \mathop \in \R}$
Then:
- $\bigcup \SS = \R$.
Finite Subfamily of Unbounded Above Open Real Intervals
Let $\R$ denote the set of real numbers.
For a given $a \in \R$, let $S_a$ denote the (real) interval:
- $S_a = \openint a \to = \set {x \in \R: x > a}$
Let $\SS$ denote the family of sets indexed by $\R$:
- $\SS := \family {S_a}_{a \mathop \in \R}$
Let $\TT$ be a finite subfamily of $\SS$.
Then:
- $\bigcup \TT$ is a proper subset of $\R$.
Also see
- Union of Doubleton for a proof that $\bigcup \set {S, T} = S \cup T$
- Results about set union can be found here.
Sources
- 1951: J.C. Burkill: The Lebesgue Integral ... (previous) ... (next): Chapter $\text {I}$: Sets of Points: $1 \cdot 1$. The algebra of sets
- 1951: Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts ... (previous) ... (next): Introduction $\S 1$: Operations on Sets
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 4$: Unions and Intersections
- 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 1.8$. Sets of sets
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 3$: Unions and Intersections of Sets
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 0.2$. Sets
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Unions and Intersections
- 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.8$: Collections of Sets: Definition $8.1$
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 5.5$, $\S 5.7$
- 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
- 1999: András Hajnal and Peter Hamburger: Set Theory ... (previous) ... (next): $1$. Notation, Conventions: $8$
- 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: Operations on Sets