Definition:Set/Implicit Set Definition

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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


$(2): \quad$ 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.


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