Definition:Set/Implicit Set Definition/Infinite Set
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Definition
If the elements in a set have an obvious pattern to them, we can define the set implicitly by using an ellipsis ($\ldots$).
If there is no end to the list of elements in the set, the ellipsis can be left open:
- $S = \set {1, 2, 3, \ldots}$
which is taken to mean:
- $S = $ the set containing $1, 2, 3, $ and so on for ever.
Also see
Sources
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 0.2$. Sets
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Introduction: Set-Theoretic Notation
- 1968: A.N. Kolmogorov and S.V. Fomin: Introductory Real Analysis ... (previous) ... (next): $1$ Set Theory: $1$. Sets and Functions: $1.1$: Basic definitions
- 1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.3$: Notation for Sets
- 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): Appendix $\text A$: Sets and Functions: $\text{A}.1$: Sets
- 1999: András Hajnal and Peter Hamburger: Set Theory ... (previous) ... (next): $1$. Notation, Conventions: $5$