Definition:Set Union/Family of Sets/Two Sets
Jump to navigation
Jump to search
Definition
Let $I = \set {\alpha, \beta}$ be an indexing set containing exactly two elements.
Let $\family {S_i}_{i \mathop \in I}$ be a family of sets indexed by $I$.
From the definition of the union of $S_i$:
- $\ds \bigcup_{i \mathop \in I} S_i := \set {x: \exists i \in I: x \in S_i}$
it follows that:
- $\ds \bigcup \set {S_\alpha, S_\beta} := S_\alpha \cup S_\beta$
Sources
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 9$: Families
- 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
- 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
- 1999: András Hajnal and Peter Hamburger: Set Theory ... (previous) ... (next): $1$. Notation, Conventions: $12$