---- > [!definition] Definition. ([[disjoint union]]) > The **disjoint union** of an family $(A_{i} : i \in I)$ of sets indexed by $I$ is the set formed from the elements of the $A_{i}$, but with each element 'remembering where it came from': $\coprod_{i \in I}A_{i}:= \bigcup_{i \in I}^{} \{ (x,i) : x \in A_{i} \}$ > ![[CleanShot 2024-05-16 at 10.57.20.jpg|400]] > From Wikipedia. > ^definition > [!generalization] > - See [[categorical coproduct]] along with [[coproducts are disjoint unions in set]] > [!specialization] > - In the special case where $X$ is a set containing each $A_{i}$, and $A_{j} \cap A_{k} = \emptyset$ for all $j,k \in I$, then we don't need to track indices in order to keep the $A_{i}$ separate—meaning that there is a [[bijection|bijective]] correspondence$\bigsqcup_{i \in I}^{} A_{i} = \bigcup_{i \in I}^{} \{ (x, i) : x \in A_{i} \} \leftrightarrow \bigcup_{i \in I}^{} \{ x : x \in A_{i} \}=\bigcup_{i \in I}^{} A_{i}.$This justifies the use of the notation $\bigsqcup_{i \in I}A_{i}$ to denote the union of the $A_{i}$ *as subsets of $X$*. > - In the special case where each $A_{i}$ equals the same set $A$, we have $\bigsqcup_{i \in I} A_{i}=\bigcup_{i \in I}^{}\{ (x, i) : x \in A\}= \{ (x, i) : x \in A, i \in I \}=A \times I.$ ^specialization ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` #reformatrevisebatch03