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 non-empty. 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 uncountable¹.

¹ The concept of countable/uncountable will be introduced, when we will be studying cardinal numbers.

| | | | | created: 2019-09-08 10:17:36 | modified: 2019-09-08 10:25:02 | by: bookofproofs | references: [8297]