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## Definition of finiteness

We define a set to be finite if and only either
• is empty, in which case we say that the set has zero elements,
or
• there is a bijection of with for some positive integer , in which case we say that has elements.
Using this language, we have proved above:
• (E)
If is a non-empty set and there is an injection , for some positive integer , then is a finite set.
Also either is a bijection or and has fewer than elements.
• (F)
If is a non-empty set and there is a surjection , for some positive integer , then is a finite set.
Also either is a bijection or and has fewer than elements.
The following corollaries follow immediately:
• (G)
If is an injection of non-empty sets, then if is finite, so is .
Also either is a bijection, or the number of elements of is less than that of .
• (H)
If is an surjection of non-empty sets, then if is finite, so is .
Also either is a bijection, or the number of elements of is less than that of .

Next: The Pigeonhole Principle, injective Up: Finiteness09february2012 Previous: Surjections from
George A. J. Sparling 2012-02-09