TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a disjoint union (or discriminated union) of a family $\left(A_i : i\in I\right)$ of sets is a set $A,$ often denoted by $\bigsqcup_ A_i,$ with an
injective function In , an injective function (also known as injection, or one-to-one function) is a that maps elements to distinct elements; that is, implies . In other words, every element of the function's is the of one element of its . The term must no ...

of each $A_i$ into $A,$ such that the images of these injections form a partition of $A$ (that is, each element of $A$ belongs to exactly one of these images). The disjoint union of a family of pairwise disjoint sets is their union. In terms of
category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
, the disjoint union is the
coproduct In category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled d ...
of the
category of sets In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
. The disjoint union is thus defined
up to Two mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
a bijection. A standard way for building the disjoint union is to define $A$ as the set of
ordered pair In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

s $\left(x, i\right)$ such that $x \in A_i,$ and the injective functions $A_i \to A$ by $x \mapsto \left(x, i\right).$

# Example

Consider the sets $A_0 = \$ and $A_1 = \.$ It is possible to index the set elements according to set origin by forming the associated sets $\begin A^*_0 & = \ \\ A^*_1 & = \, \\ \end$ where the second element in each pair matches the subscript of the origin set (for example, the $0$ in $\left(5, 0\right)$ matches the subscript in $A_0,$ etc.). The disjoint union $A_0 \sqcup A_1$ can then be calculated as follows: $A_0 \sqcup A_1 = A^*_0 \cup A^*_1 = \.$

# Set theory definition

Formally, let $\left\$ be a
family of sets In set theory Set theory is the branch of that studies , which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of , is mostly concerned with those that are ...
indexed by $I.$ The disjoint union of this family is the set $\bigsqcup_ A_i = \bigcup_ \left\.$ The elements of the disjoint union are ordered pairs $\left(x, i\right).$ Here $i$ serves as an auxiliary index that indicates which $A_i$ the element $x$ came from. Each of the sets $A_i$ is canonically isomorphic to the set $A_i^* = \left\.$ Through this isomorphism, one may consider that $A_i$ is canonically embedded in the disjoint union. For $i \neq j,$ the sets $A_i^*$ and $A_j^*$ are disjoint even if the sets $A_i$ and $A_j$ are not. In the extreme case where each of the $A_i$ is equal to some fixed set $A$ for each $i \in I,$ the disjoint union is the
Cartesian product In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
of $A$ and $I$: $\bigsqcup_ A_i = A \times I.$ Occasionally, the notation $\sum_ A_i$ is used for the disjoint union of a family of sets, or the notation $A + B$ for the disjoint union of two sets. This notation is meant to be suggestive of the fact that the
cardinality In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of the disjoint union is the of the cardinalities of the terms in the family. Compare this to the notation for the
Cartesian product In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
of a family of sets. Disjoint unions are also sometimes written $\biguplus_ A_i$ or $\ \cdot\!\!\!\!\!\bigcup_ A_i.$ In the language of
category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
, the disjoint union is the
coproduct In category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled d ...
in the
category of sets In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
. It therefore satisfies the associated
universal property In category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), ...
. This also means that the disjoint union is the
categorical dual In category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled di ...
of the
Cartesian product In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
construction. See
coproduct In category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled d ...
for more details. For many purposes, the particular choice of auxiliary index is unimportant, and in a simplifying
abuse of notation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, the indexed family can be treated simply as a collection of sets. In this case $A_i^*$ is referred to as a of $A_i$ and the notation $\underset A$ is sometimes used.

# Category theory point of view

In
category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
the disjoint union is defined as a
coproduct In category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled d ...
in the category of sets. As such, the disjoint union is defined up to an isomorphism, and the above definition is just one realization of the coproduct, among others. When the sets are pairwise disjoint, the usual union is another realization of the coproduct. This justifies the second definition in the lead. This categorical aspect of the disjoint union explains why $\coprod$ is frequently used, instead of $\bigsqcup,$ to denote ''coproduct''.