# Union symbol in LaTeX

In mathematics, given two sets A and B, we define their union as the set formed by elements both from A and B. This new set is denoted by A B, and the symbol used for it can be produced with the command \cup inside math mode. Using set notation, the rigorous definition of the union of sets is:

$$A\cup B=\{x\,\mid\, x \in A \text{ or } x \in B\}$$

This output was produced by the code:

% Union symbol in LaTeX
$A\cup B=\{x\,\mid\, x \in A \text{ or } x \in B\}$

## Big union symbol

But this is not the end of the story. There are other symbols in mathematics that are necessary to denote unions. For example, when we have a family of set $$\{A_i\}_{i \in I}$$ indexed on another set I (that is, the elements of I act as labels, and we have a set for each of these labels) then the union of this sets is denoted with the big union symbol:

$$\displaystyle \bigcup_{i \in I} A_i$$

In fact, the rigorous definition of this set is:

$$\displaystyle \bigcup_{i \in I} A_i=\{x\,\mid\,x \in A_i \text{ for some } i \in I\}$$

This big union is produced with the command \bigcup, and the contents below it are passed as a subindex so that the line of code inside math mode used to produce this last equation was:

% Big Union symbol in LaTeX
$\bigcup_{i \in I} A_i=\{x\,\mid\,x \in A_i \text{ for some } i \in I\}$

## Disjoint union symbol in LaTeX

We say that a union of sets is disjoint when any pair of sets from the family that constitutes the union share any common element. Using the intersection of sets (defined as the set of common elements between two sets) a union $$A \cup B$$ is disjoint if $$A\cap B=\emptyset$$, where $$\emptyset$$ denotes the empty set, that is, the set with no elements. Sometimes the special symbol $$\sqcup$$ is used to denote the disjoint union, $$A \sqcup B$$. This symbol is produced with the command \sqcup.

Other symbols such as $$\uplus$$ and $$\dot{\cup}$$ are also used, and they are produced with the commands \uplus and \dot{\cup}. All of them have their corresponding “big” version, to have an index below them in displayed math mode. Finally, in category theory, the disjoint union is denoted using the symbol $$\coprod$$, which is produced with the command \coprod.

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