Cardinality

The cardinality of the set of integers is according to \aleph_0 (aleph-null). This is readily approved by the architecture of a bijection, that is, a action that is injective and surjective from Z to N.citation needed

If N = {0, 1, 2, ...} again accede the function:

f(x) = \begin{cases} 2|x|, & \mbox{if } x < 0 \\ 0, & \mbox{if } x = 0 \\ 2x-1, & \mbox{if } x > 0. \end{cases}

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

If N = {1,2,3,...} again accede the function:

g(x) = \begin{cases} 2|x|, & \mbox{if } x < 0 \\ 2x+1, & \mbox{if } x \ge 0. \end{cases}

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

If the area is belted to Z again anniversary and every affiliate of Z has one and alone one agnate affiliate of N and by the analogue of basal adequation the two sets accept according cardinality.citation neede