Definition:Disjoint Union (Set Theory)/2 Sets
Definition
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 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
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 5$. Induced mappings; composition; injections; surjections; bijections: Exercise $18$
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): disjoint union
- 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $2$: Partial Order Relations