Definition:Set/Implicit Set Definition
Definition
If the elements in a set have an obvious pattern to them, we can define the set implicitly by using an ellipsis ($\ldots$).
For example, suppose $S = \set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}$.
A more compact way of defining this set is:
- $S = \set {1, 2, \ldots, 10}$
With this notation we are asked to suppose that the numbers count up uniformly, and we can read this definition as:
- $S$ is the set containing $1$, $2$, and so on, up to $10$.
See how this notation is used: there is a comma before the ellipsis and one after it. It is a very good idea to be careful with this.
The point needs to be made: "how obvious is obvious?"
If there is any doubt as to the precise interpretation of an ellipsis, either the set should be defined by predicate, or explicit definition should be used.
Infinite Set
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.
Multipart Infinite Set
Let $S$ be a set.
Suppose $S$ is to contain:
- $(1): \quad$ a never-ending list of elements
and
- $(2): \quad$ other elements which are unrelated to that list (perhaps another never-ending list).
Then a semicolon is used to separate the various conceptual parts:
- $S = \set {1, 3, 5, \ldots; 2, 4, 6, \ldots; \text{red}, \text{orange}, \text{green} }$
Note that without the semicolon it would appear as though the first list (of odd numbers) continued as the second list (of even numbers) which in turn continued as a list of colours, which is absurd.
Examples
Letters of the Alphabet
An example in natural language of implicit set definition is:
- $S := \set {A, B, C, D, \dotsc, Z}$
That is, $S$ is the set of capital letters of the alphabet.
This definition is actually ambiguous, as it is not made clear exactly which alphabet is under consideration here.
The English one is assumed by context.
Also see
Sources
- 1963: George F. Simmons: Introduction to Topology and Modern Analysis ... (previous) ... (next): $\S 1$: Sets and Set Inclusion
- 1965: A.M. Arthurs: Probability Theory ... (previous) ... (next): Chapter $1$: Set Theory: $1.2$: Sets and subsets
- 1965: Claude Berge and A. Ghouila-Houri: Programming, Games and Transportation Networks ... (previous) ... (next): $1$. Preliminary ideas; sets, vector spaces: $1.1$. Sets
- 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
- 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): Chapter $1$: Sets, Functions, and Relations: $\S 1$: Sets and Membership
- 1971: Patrick J. Murphy and Albert F. Kempf: The New Mathematics Made Simple (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 2$: Sets and functions: Sets
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 2$: Sets and Subsets
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): Notation and Terminology
- 1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): Appendix $\text{A}.1$: Sets and Subsets
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 2$: Introductory remarks on sets: $\text{(d)}$
- 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: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Basic Notations
- 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: $5$
- 2008: David Joyner: Adventures in Group Theory (2nd ed.) ... (previous) ... (next): Chapter $1$: Elementary, my dear Watson: $\S 1.2$: Elements, my dear Watson