Countable set
A set is countable if it is finite or there is a bijection : its elements can be listed as .
Listing the elements
Countability means that the elements of can be listed:
where the list may or may not terminate: it terminates exactly when is finite, and otherwise is countably infinite.
Examples
-
Any finite set is countable by definition.
-
is countable.
-
is countable: list the integers as
so a bijection is given by
Size interpretation
A set is countable precisely when it is at most as big as . Accordingly:
- an injection makes countable;
- a surjection , or , likewise makes countable.
Related
Stated in
- Definition 6.1 (Countable Set)§6 Countability