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Example

• The set S := {0, 2, 4} is finite; its size is 3 because we can define a function f: N₃ ->^bijective S as

  f(0)	:=	0
  f(1)	:=	2
  f(2)	:=	4

  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 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.