◀ ▲ ▶Branches / Settheory / Definition: Generalized Union of Sets
The set union is defined for only two sets. Sometimes, it is convenient to have a more general definition involving an arbitrary number of sets.
Definition: Generalized Union of Sets
Let $X_i\text{ , }i\in I$ be a family of sets over the index set $I$. A union of sets of $X_i\text{ , }i\in I$ is denoted and defined by $$\bigcup_{i\in I}X_i:=\{x\in X\mid \exists i\in I\text{, }x\in X_i\}.$$
Notes
 The union of sets is a generalized case of the set union $A\cup B,$ since if $U$ is a universal set of $A$ and $B,$ then the index set consists only of two elements, e.g. $I=\{1,2\},$ $A=U_1$, $B=U_2$ and $$A\cup B=\{x\in U\mid \exists i\in \{1,2\}\text{, }x\in U_i\}=\{x\in U\mid x\in A\vee x\in B\}.$$
 The only thing which is required for the index set $I$ of an index family is that it is nonempty. It can be any set, in particular, it does not have to consist of the commonly used positive integers. You can have any kind of indices, even indices which are uncountable^{1}.
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References
Bibliography
 Flachsmeyer, Jürgen: "Kombinatorik", VEB Deutscher Verlag der Wissenschaften, 1972, 3rd Edition
Footnotes