Definition:Set Union

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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 $S \cup T$:

VennDiagramSetUnion.png


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


  • 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

  1. See Earliest Uses of Symbols of Set Theory and Logic in Jeff Miller's website Earliest Uses of Various Mathematical Symbols.


Sources