# 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:

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

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 $I$:

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

### $2$ Sets

Let $I$ be a doubleton, say $I := \set {0, 1}$.

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

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

\(\ds \bigsqcup_{i \mathop \in I} S_i\) | \(=\) | \(\ds \bigcup_{i \mathop \in I} \set {\tuple {x, 0}, \tuple {y, 1} : x \in S_0, y \in S_1}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \bigcup \set {S_0 \times \set 0, S_1 \times \set 1}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \paren {S_0 \times \set 0} \cup \paren {S_1 \times \set 1}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds S_0 \sqcup S_1\) |

### Disjoint Sets

Let $A$ and $B$ be disjoint sets, that is:

- $A \cap B = \O$

Then the disjoint union of $A$ and $B$ can be defined as:

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

where $A \cup B$ is the (usual) set union of $A$ and $B$.

## Also known as

A **disjoint union** in the context of set theory 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.*

## Notation

The notations:

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

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

When two sets are under consideration, the notation:

- $A \sqcup B$

or:

- $A \coprod B$

are usually used.

Some sources use:

- $A \vee B$

The notations:

- $A + B$

or

- $A \oplus B$

are also encountered sometimes.

This notation reflects the fact that, from the Cardinality of Set Union: Corollary, 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 - 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*... (previous) ... (next): Appendix $\text A$: Sets and Functions: $\text{A}.2$: Boolean Operations