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 Set
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.

Also see

 * Definition:Explicit Set Definition
 * Definition:Set Definition by Predicate