Definition:Set Union

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Definition

Let $S$ and $T$ be any two sets.


The union (or logical sum, or sum) 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 $\cup$ is an operator.


Generalized Notation

Let $I$ be an indexing set.

Let $\left \langle {X_i} \right \rangle_{i \in I}$ be a family of subsets of a set $S$.

Then the union of $\left \langle {X_i} \right \rangle$ is defined as:

$\displaystyle \bigcup_{i \in I} X_i = \left\{{y: \exists i \in I: y \in X_i}\right\}$

If the indexing set is clear from context, the notation $\displaystyle \bigcup_i X_i$ can be used.


The indexing set itself can be disposed of, as follows:

If $\mathbb S$ is a set of sets, then 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:

$\displaystyle S \cup T = \bigcup \left\{{S, T}\right\}$


Countable Union

In the general definition of union, if the indexing set $I$ or the set of sets $\mathbb S$ is countable, then the term countable union is used.

The union of $X_1, X_2, X_3, \ldots$ is denoted:

$X_1 \cup X_2 \cup X_3 \cup \cdots$
$\displaystyle \bigcup_{k=1}^\infty X_k$
$\displaystyle \bigcup_{k \in \N} X_k$

The first of these is colloquial and should be avoided.


Finite Union

In the general definition of union, if the indexing set $I$ or the set of sets $\mathbb S$ is finite, then the term finite union is used.

The union of $X_1, X_2, \ldots, X_n$ is denoted:

$X_1 \cup X_2 \cup \cdots \cup X_n$
$\displaystyle \bigcup_{k=1}^n X_k$
$\displaystyle \bigcup_{1 \le k \le n} X_k$


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 Unions in Zermelo-Fraenkel set theory:

$\forall A: \exists x: \forall y: \left({y \in x \iff \exists z: \left({z \in A \land y \in z}\right)}\right)$


Notation

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.


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 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

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