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- The set S := {0, 2, 4} is finite; its size is 3 because we can
define a function
`f`:**N**_{3}->^{bijective}`S`as

i.e., f = [0, 2, 4]. The length of f is the same as the length of [0, 4, 2], [4, 2, 0] or of any other bijection to`f`(0):= 0 `f`(1):= 2 `f`(2):= 4 `S`. - The set
**N**is infinite. If it were finite, we had some`n`in**N**and some f:**N**_{n}->^{bijective}**N**. Take`k`:= 1+max{`f`(`i`):`i`in**N**_{n}}. Then`k`in**N**but**forall**i in**N**_{n}:`f`(i) !=`k`, i.e.,`f`is not surjective on**N**.

Author: Wolfgang Schreiner

Last Modification: December 7, 1999