Definition:Set Union
Contents
Definition
Let $S$ and $T$ be any two sets.
The (set) union of $S$ and $T$ is written $S \cup T$.
It means the set which consists of all the elements which are contained in either (or both) of $S$ and $T$:
- $x \in S \cup T \iff x \in S \lor x \in T$
or, slightly more formally:
- $A = S \cup T \iff \forall z: \left({z \in A \iff z \in S \lor z \in T}\right)$
We can write:
- $S \cup T := \left\{{x: x \in S \lor x \in T}\right\}$
For example, let $S = \left \{{1,2,3}\right\}$ and $T = \left \{{2,3,4}\right\}$. Then $S \cup T = \left \{{1,2,3,4}\right\}$.
It can be seen that, in this form, $\cup$ is a binary operation which acts on sets.
General Definition
Let $\mathbb S$ be a set of sets.
The union of $\mathbb S$ is:
- $\displaystyle \bigcup \mathbb S := \left\{{x: \exists X \in \mathbb S: x \in X}\right\}$
That is, the set of all elements of all elements of $\mathbb S$.
Thus the general union of two sets can be defined as:
- $\displaystyle \bigcup \left\{{S, T}\right\} = S \cup T$
Family of Sets
Let $I$ be an indexing set.
Let $\left \langle {S_i} \right \rangle_{i \mathop \in I}$ be a family of sets indexed by $I$.
Then the union of $\left \langle {S_i} \right \rangle$ is defined as:
- $\displaystyle \bigcup_{i \mathop \in I} S_i := \left\{{x: \exists i \in I: x \in S_i}\right\}$
Countable Union
Let $\mathbb S$ be a set of sets.
Let $\left\langle{S_n}\right\rangle_{n \in \N}$ be a sequence in $\mathbb S$.
Let $S$ be the union of $\left\langle{S_n}\right\rangle_{n \in \N}$:
- $\displaystyle S = \bigcup_{n \mathop \in \N} S_n$
Then $S$ is a countable union of sets in $\mathbb S$.
It can also be denoted:
- $\displaystyle S = \bigcup_{n \mathop = 0}^\infty S_n$
but its usage is strongly discouraged.
If there is no danger of ambiguity, we can also write:
- $\displaystyle S = \bigcup_{\N} S_n$.
Finite Union
Let $S = S_1 \cup S_2 \cup \ldots \cup S_n$.
Then:
- $\displaystyle S = \bigcup_{i \mathop \in \N^*_n} S_i = \left\{{x: \exists i \in \N^*_n: x \in S_i}\right\}$
where $\N^*_n = \left\{{1, 2, 3, \ldots, n}\right\}$.
If it is clear from the context that $i \in \N^*_n$, we can also write $\displaystyle \bigcup_{\N^*_n} S_i$.
Illustration by Venn Diagram
The red area in the following Venn diagram illustrates $S \cup T$:
Axiomatic Set Theory
The concept of set union is axiomatised in the Axiom of Union in Zermelo-Fraenkel set theory:
- $\forall A: \exists x: \forall y: \left({y \in x \iff \exists z: \left({z \in A \land y \in z}\right)}\right)$
Historical Note
The symbol $\cup$, informally known as cup, was first used by Hermann Grassmann in Die Ausdehnungslehre from 1844. However, he was using it as a general operation symbol, not specialized for union.
It was Giuseppe Peano who took this symbol and used it for union, in his 1888 work Calcolo geometrico secondo l'Ausdehnungslehre di H. Grassmann.
Peano also created the large symbol $\bigcup$ for general union of more than two sets. This appeared in his Formulario Mathematico (5th edtion, 1908).^{[1]}
Also known as
The union of sets is also known as the logical sum, or just sum, but these terms are usually considered old-fashioned nowadays.
Some authors use the notation $S + T$ for $S \cup T$, but this is non-standard and can be confusing, so its use is not recommended.
Also, $S + T$ is sometimes used for disjoint union.
Also see
- Definition:Set Intersection, a related operation.
- Union of Singleton, where it is shown that $\displaystyle \mathbb S = \left\{{S}\right\} \implies \bigcup \mathbb S = S$
- Union of Empty Set, where it is shown that $\displaystyle \mathbb S = \varnothing \implies \bigcup \mathbb S = \varnothing$
- Results about set unions can be found here.
Internationalization
Union is translated:
In French: | somme | (literally: sum) |
References
- ↑ See Earliest Uses of Symbols of Set Theory and Logic in Jeff Miller's website Earliest Uses of Various Mathematical Symbols.
Sources
- Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 4$: Unions and Intersections
- W.E. Deskins: Abstract Algebra (1964)... (previous)... (next): $\S 1.1$: Definition $1.2$
- Steven A. Gaal: Point Set Topology (1964)... (previous)... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets
- Murray R. Spiegel: Theory and Problems of Complex Variables (1964)... (previous)... (next): $1$: Point Sets: $12.$
- J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 1.4$
- Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 3$
- Richard A. Dean: Elements of Abstract Algebra (1966)... (previous)... (next): $\S 0.2$
- George McCarty: Topology: An Introduction with Application to Topological Groups (1967)... (previous)... (next): Introduction: Set-Theoretic Notation
- A.N. Kolmogorov and S.V. Fomin: Introductory Real Analysis (1968)... (previous)... (next): $\S 1.2$: Operations on sets
- Ian D. Macdonald: The Theory of Groups (1968)... (previous)... (next): Appendix: Elementary set and number theory
- C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 1.2$: Example $2$
- Avner Friedman: Foundations of Modern Analysis (1970)... (previous)... (next): $\S 1.1$: Rings and Algebras
- Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 6$
- Robert H. Kasriel: Undergraduate Topology (1971)... (previous)... (next): $\S 1.4$: Union and Intersection of Sets
- Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory (1971): $\S 5.6$
- A. G. Howson: A Handbook of Terms used in Algebra and Analysis (1972)... (previous)... (next): $\S 2$: Sets and functions: Sets
- T.S. Blyth: Set Theory and Abstract Algebra (1975)... (previous)... (next): $\S 1$
- W.A. Sutherland: Introduction to Metric and Topological Spaces (1975)... (previous)... (next): Notation and Terminology
- Gary Chartrand: Introductory Graph Theory (1977)... (previous)... (next): Appendix $\text{A}.1$: Sets and Subsets
- Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 7$
- P.M. Cohn: Algebra Volume 1 (2nd ed., 1982)... (previous)... (next): $\S 1.2$: Sets
- Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction (1986)... (previous)... (next): $\S 1.2$: Outcomes and events
- Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed., 1993)... (previous)... (next): $\S 1.2$: Operations on Sets
- Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed., 1993)... (previous)... (next): $\S 1.3$: Notation for Sets: Exercise $1.3.1 \ \text{(i)}$
- H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability (1996)... (previous)... (next): Appendix $\text{A}.2$: Boolean Operations
- James R. Munkres: Topology (2nd ed., 2000)... (previous)... (next): $1$: Set Theory and Logic: $\S 1$: Fundamental Concepts
- René L. Schilling: Measures, Integrals and Martingales (2005)... (next): $\S 2$
- Paul Halmos and Steven Givant: Introduction to Boolean Algebras (2008)... (previous)... (next): Appendix $\text{A}$: Set Theory: Operations on Sets
- M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed., 2012)... (previous)... (next): Appendix $\text{A}.2$: Definition $\text{A}.8$