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 $S = \left\{{1, 2, 3, 4, 5, 6, 7, 8, 9, 10} \right\}$.

A more compact way of defining this set is:
 * $S = \left\{{1, 2, \ldots, 10}\right\}$

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: we have 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 Sets
If there is no limit to the 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.

See Infinite.

Multipart Infinite Sets
If the set is to contain a never-ending list of elements and other elements which are unrelated to that list (perhaps another never-ending list), a semicolon can be used to separate the various conceptual parts:
 * $S = \left\{{1, 3, 5, \ldots; 2, 4, 6, \ldots; \text{red}, \text{orange}, \text{green}}\right\}$

The point here is to 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.