Integer - Cardinality

Cardinality

The cardinality of the set of integers is equal to (aleph-null). This is readily demonstrated by the construction of a bijection, that is, a function that is injective and surjective from Z to N. If N = {0, 1, 2, ...} then consider the function:

{ ... (-4,8) (-3,6) (-2,4) (-1,2) (0,0) (1,1) (2,3) (3,5) ... }

If N = {1,2,3,...} then consider the function:

{ ... (-4,8) (-3,6) (-2,4) (-1,2) (0,1) (1,3) (2,5) (3,7) ... }

If the domain is restricted to Z then each and every member of Z has one and only one corresponding member of N and by the definition of cardinal equality the two sets have equal cardinality.