Definition:Set Union

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Definition

Let $S$ and $T$ be sets.


The (set) union of $S$ and $T$ is the set $S \cup T$, 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:

$A = S \cup T \iff \forall z: \paren {z \in A \iff z \in S \lor z \in T}$


We can write:

$S \cup T := \set {x: x \in S \lor x \in T}$

and can voice it $S$ union $T$.


It can be seen that, in this form, $\cup$ is a binary operation which acts on sets.


Set of Sets

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:

$\bigcup \set {S, T} = S \cup T$


Family of Sets

Let $I$ be an indexing set.

Let $\family {S_i}_{i \mathop \in I}$ be a family of sets indexed by $I$.


Then the union of $\family {S_i}$ is defined as:

$\ds \bigcup_{i \mathop \in I} S_i := \set {x: \exists i \in I: x \in S_i}$


Countable Union

Let $\mathbb S$ be a set of sets.

Let $\sequence {S_n}_{n \mathop \in \N}$ be a sequence in $\mathbb S$.

Let $S$ be the union of $\sequence {S_n}_{n \mathop \in \N}$:

$\ds S = \bigcup_{n \mathop \in \N} S_n$


Then $S$ is a countable union of sets in $\mathbb S$.


Finite Union

Let $S = S_1 \cup S_2 \cup \ldots \cup S_n$.

Then:

$\ds S = \bigcup_{i \mathop \in \N^*_n} S_i = \set {x: \exists i \in \N^*_n: x \in S_i}$

where $\N^*_n = \set {1, 2, 3, \ldots, n}$.


If it is clear from the context that $i \in \N^*_n$, we can also write $\ds \bigcup_{\N^*_n} S_i$.


Illustration by Venn Diagram

The union $S \cup T$ of the two sets $S$ and $T$ is illustrated in the following Venn diagram by the red area:

VennDiagramSetUnion.png


Axiomatic Set Theory

The concept of set union is axiomatised in the Axiom of Unions in various versions of axiomatic set theory:

For every set of sets $A$, there exists a set $x$ (the union set) that contains all and only those elements that belong to at least one of the sets in the $A$:

$\forall A: \exists x: \forall y: \paren {y \in x \iff \exists z: \paren {z \in A \land y \in z} }$


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.

They are also used to mean other concepts.

The term join can also be seen, but this is usually reserved for specific contexts.


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.


Examples

Example: $2$ Arbitrarily Chosen Sets

Let:

\(\ds S\) \(=\) \(\ds \set {a, b, c}\)
\(\ds T\) \(=\) \(\ds \set {c, e, f, b}\)

Then:

$S \cup T = \set {a, b, c, e, f}$


Example: $2$ Arbitrarily Chosen Sets of Complex Numbers

Let:

\(\ds A\) \(=\) \(\ds \set {3, -i, 4, 2 + i, 5}\)
\(\ds B\) \(=\) \(\ds \set {-i, 0, -1, 2 + i}\)

Then:

$A \cup B = \set {3, -i, 0, -1, 4, 2 + i, 5}$


Example: $3$ Arbitrarily Chosen Sets

Let:

\(\ds A_1\) \(=\) \(\ds \set {1, 2, 3, 4}\)
\(\ds A_2\) \(=\) \(\ds \set {1, 2, 5}\)
\(\ds A_3\) \(=\) \(\ds \set {2, 4, 6, 8, 12}\)

Then:

$A_1 \cup A_2 \cup A_3 = \set {1, 2, 3, 4, 5, 6, 8, 12}$


Example: $3$ Arbitrarily Chosen Sets of Complex Numbers

Let:

\(\ds A\) \(=\) \(\ds \set {1, i, -i}\)
\(\ds B\) \(=\) \(\ds \set {2, 1, -i}\)
\(\ds C\) \(=\) \(\ds \set {i, -1, 1 + i}\)

Then:

$\paren {A \cup B} \cup C = \set {2, 1, -i,1, 1 + i}$


Example: People who are Blue-Eyed or British

Let:

\(\ds B\) \(=\) \(\ds \set {\text {British people} }\)
\(\ds C\) \(=\) \(\ds \set {\text {Blue-eyed people} }\)

Then:

$B \cup C = \set {\text {People who are blue-eyed or British or both} }$


Example: Overlapping Integer Sets

Let:

\(\ds A\) \(=\) \(\ds \set {x \in \Z: 2 \le x}\)
\(\ds B\) \(=\) \(\ds \set {x \in \Z: x \le 5}\)

Then:

$A \cup B = \Z$


Example: Subset of Union

Let $U, V, W$ be non-empty sets.

Let $W$ be such that for all $w \in W$, either:

$w \in U$

or:

$w \in V$

Then:

$W \subseteq U \cup V$


Also see


  • Union of Singleton, where it is shown that $\mathbb S = \set S \implies \bigcup \mathbb S = S$
  • Union of Empty Set, where it is shown that $\mathbb S = \O \implies \bigcup \mathbb S = \O$
  • Results about set unions can be found here.


Internationalization

Union is translated:

In French: somme  (literally: sum)
In French: union
In French: réunion
In Dutch: vereniging


Historical Note

The concept of set union, or logical addition, was stated by Leibniz in his initial conception of symbolic logic.


The symbol $\cup$, informally known as cup, was first used by Hermann Günter Grassmann in Die Ausdehnungslehre from $1844$.

However, he was using it as a general operation symbol, not specialized for set 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 ed. of $1908$.


Sources

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