nLab
disjoint union

Idea

The disjoint union is a coproduct in Set, the category of sets.

In a general category coproducts need not have the expected disjointness property of those in Set. If they do they are called disjoint coproducts.

Definition

Given any family $(A_i{)}_{i:I}$ of sets, the (external) disjoint union ${⨄}_i{A}_i$ (also written ${\sum }_i{A}_i$, ${\coprod }_i{A}_i$, etc) of the family is the set of all (ordered) pairs $(i,a)$ with $i$ in the index set $I$ and $a$ in $A_i$.

As stated, the second element of such a pair depends on the first element, which is natural in dependent type theory and certainly feasible in material set theory; but if you don’t like this, then define ${⨄}_i{A}_i$ to be the set of those elements $x$ of the cartesian product ${\prod }_i𝒫A_i$ of the power sets such that there is exactly one index $j$ such that $x_j$ is inhabited and that $x_j$ is a singleton. If you're trying to be predicative too, consider adopting the existence of disjoint unions as an axiom (the axiom of disjoint unions) in your foundations.

There is a natural injection $A_j\to {⨄}_i{A}_i$ (mapping $a$ to $(j,a)$) for each index $j$, and its common to treat $A_j$ as a subset of ${⨄}_i{A}_i$. So if no confusion can result (in particular, when the notation for an elements of $A_j$ always makes the ambient set clear), one often suppresses the index in the notation for an element of the disjoint union.

Special cases

Given sets $A$ and $B$, the disjoint union of the binary family $(A,B)$ is written $A\uplus B$ (also $A+B$, $A⨿B$, etc); its elements may be written (if care is needed) as $(0,a)$ and $(1,b)$, $(1,a)$ and $(2,b)$, $\iota a$ and $\kappa b$, and in many other styles.

Given sets $A_1$ through $A_n$, the disjoint union of the $n$-ary family $(A_1,\dots ,A_n)$ is written ${⨄}_{i=1}^n{A}_i$ (or similarly); its elements may be written (if care is needed) as $(i,a)$ for $1\le i\le n$ and $a\in A_i$.

Given sets $A_1$, $A_2$, etc, the disjoint union of the countably infinitary family $(A_1,A_2,\dots )$ is written ${⨄}_{i=1}^{\mathrm{\infty }}{A}_i$ (or similarly); its elements may be written (if care is needed) as $(i,a)$ for $i$ a natural number and $a\in A_i$.

Given a set $A$, the disjoint union of the unary family $(A)$ may be identified with $A$ itself; that is, we identify $(i,a)$ for the unique index $i$ with $a$.

The disjoint union of the empty family $()$ is empty; it has no elements.

Internal version

(This is internal in the sense of ‘internal direct sum’, not internalization. For that, just see coproduct.)

If a family $(A_i{)}_{i:I}$ of subsets of a given set $X$ are all pairwise disjoint (that is, $A_i\cap A_j$ has an element only if $i=j$, for any indices $i$ and $j$), then the union ${\bigcup }_i{A}_i$ is naturally bijective with the (external) disjoint union defined above. Conversely, given an external disjoint union ${⨄}_i{A}_i$, each $A_j$ may be identified with a subset of ${⨄}_i{A}_i$ (as explained above); these subsets are all pairwise disjoint, and their union is the entire disjoint union.

Accordingly, a union of pairwise disjoint subsets may be called an internal disjoint union. (Compare the internal vs external notions of direct sum.)

Related concepts

• union

category: foundational axiom