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