Definition:Set Union/Family of Sets/Subsets of General Set

Definition

Let $I$ be an indexing set.

Let $\family {S_i}_{i \mathop \in I}$ be an indexed family of subsets of a set $X$.

Then the union of $\family {S_i}$ is defined as:

$\ds \bigcup_{i \mathop \in I} S_i := \set {x \in X: \exists i \in I: x \in S_i}$

where $i$ is a dummy variable.

Also denoted as

The set $\ds \bigcup_{i \mathop \in I} S_i$ can also be seen denoted as:

$\ds \bigcup_I S_i$

or, if the indexing set is clear from context:

$\ds \bigcup_i S_i$

The form:

$\ds \bigcup_{S \mathop \in X} S$

can also be seen, but this obscures the true nature of the indexing set.

On $\mathsf{Pr} \infty \mathsf{fWiki}$ it is recommended that the full form is used.

Sources

• 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 9$: Families
• 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 2$: Sets and functions: Sets
• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 6$. Indexed families; partitions; equivalence relations
• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 4$: Indexed Families of Sets
• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): Notation and Terminology
• 1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.4$: Sets of Sets
• 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): Appendix $\text{A}$: Set Theory: Families