Definition
A set S is called countable if there exists an injective function f from S to the natural numbers
If f is also surjective and therefore bijective (since f is already defined to be injective), then S is called countably infinite.
As noted above, this terminology is not universal: Some authors use countable to mean what is here called "countably infinite," and to not include finite sets.
For alternative (equivalent) formulations of the definition in terms of a bijective function or a surjective function, see the section Formal definition and properties below.
Read more about this topic: Countable Set
Famous quotes containing the word definition:
“The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
—Jean Baudrillard (b. 1929)
“One definition of man is “an intelligence served by organs.””
—Ralph Waldo Emerson (1803–1882)
“It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possess—after many mysteries—what one loves.”
—François, Duc De La Rochefoucauld (1613–1680)