# Definition:Set/Implicit Set Definition

## Contents

## 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:

- $S = \left\{{1, 2, 3, \ldots}\right\}$

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:

- $S = \left\{{1, 3, 5, \ldots; 2, 4, 6, \ldots; \text{red}, \text{orange}, \text{green}}\right\}$

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

- 1962: Bert Mendelson:
*Introduction to Topology*... (previous) ... (next): $\S 1.2$: Sets and Subsets - 1966: Richard A. Dean:
*Elements of Abstract Algebra*... (previous) ... (next): $\S 0.2$ - 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): $\S 1.1$: Basic definitions - 1971: Robert H. Kasriel:
*Undergraduate Topology*... (previous) ... (next): $\S1.1$: Sets and Membership - 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: $\text{(d)}$ - 1983: George F. Simmons:
*Introduction to Topology and Modern Analysis*... (previous) ... (next): $\S 1$: Sets and Set Inclusion - 1993: Keith Devlin:
*The Joy of Sets: Fundamentals of Contemporary Set Theory*(2nd ed.) ... (previous) ... (next): $\S 1.3$: Notation for Sets - 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*... (previous) ... (next): Appendix $\text{A}.1$: 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): $\S 1.2$: Elements, my dear Watson