# Definition:Disjoint Union (Set Theory)

## Definition

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

The **disjoint union** of $\family {S_i}_{i \mathop \in I}$ is defined as the set:

- $\displaystyle \bigsqcup_{i \mathop \in I} S_i = \bigcup_{i \mathop \in I} \set {\tuple {x, i}: x \in S_i}$

where $\bigcup$ denotes union.

Each of the sets $S_i$ is canonically embedded in the **disjoint union** as the set:

- $S_i^* = \set {\tuple {x, i}: x \in S_i}$

For distinct $i, j \in I$, the sets $S_i^*$ and $S_j^*$ are disjoint even if $S_i$ and $S_j$ are not.

If $S$ is a set such that $\forall i \in I: S_i = S$, then the **disjoint union** (as defined above) is equal to the cartesian product of $S$ and $I$:

- $\displaystyle \bigsqcup_{i \mathop \in I} S = S \times I$

Where $A \cap B = \O$, we can define:

- $A \sqcup B := A \cup B$

where $A \cup B$ is the union of $A$ and $B$.

## Also known as

This is also called a **discriminated union**.

In Georg Cantor's original words:

*We denote the uniting of many aggregates $M, N, P, \ldots$, which have no common elements, into a single aggregate by*- $\tuple {M, N, P, \ldots}$.

*The elements in this aggregate are, therefore, the elements of $M$, of $N$, of $P$, $\ldots$, taken together.*

Occasionally the notations:

- $\displaystyle \sum_{i \mathop \in I} S_i$ and $\displaystyle \coprod_{i \mathop \in I} S_i$

can be seen for the **disjoint union** of a family of sets.

When two sets are under consideration, the notation:

- $A \sqcup B$

is usually used.

Some sources use:

- $A \vee B$

The notation:

- $A + B$

is also encountered sometimes.

This notation reflects the fact that, from the corollary to Cardinality of Set Union, the cardinality of the **disjoint union** is the sum of the cardinalities of the terms in the family.

It is motivated by the notation for a coproduct in category theory, combined with Disjoint Union is Coproduct in Category of Sets.

Compare this to the notation for the cartesian product of a family of sets.

## Also see

- Results about
**disjoint unions**can be found here.

## Sources

- 1915: Georg Cantor:
*Contributions to the Founding of the Theory of Transfinite Numbers*... (previous) ... (next): First Article: $\S 1$: The Conception of Power or Cardinal Number: $(2)$ - 1970: Avner Friedman:
*Foundations of Modern Analysis*... (previous) ... (next): $\S 1.1$: Rings and Algebras - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 5$. Induced mappings; composition; injections; surjections; bijections: Exercise $18$ - 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction: Functions - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**disjoint union** - 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*... (previous) ... (next): Appendix $\text A$: Sets and Functions: $\text{A}.2$: Boolean Operations