Cardinality

Two sets A and B have the same cardinality if there exists a bijection (a.k.a., one-to-one correspondence) from A to B,[10] that is, a function from A to B that is both injective and surjective. Such sets are said to be equipotent, equipollent, or equinumerous. This relationship can also be denoted A ≈ B or A ~ B.

For example, the set E = {0, 2, 4, 6, ...} of non-negative even numbers has the same cardinality as the set N = {0, 1, 2, 3, ...} of natural numbers, since the function f(n) = 2n is a bijection from N to E (see picture).

For finite sets A and B, if some bijection exists from A to B, then each injective or surjective function from A to B is a bijection. This is no longer true for infinite A and B. For example, the function g from N to E, defined by g(n) = 4n is injective, but not surjective, and h from N to E, defined by h(n) = n - (n mod 2) is surjective, but not injective. Neither g nor h can challenge |E| = |N|, which was established by the existence of f.

A has cardinality less than or equal to the cardinality of B, if there exists an injective function from A into B.

A has cardinality strictly less than the cardinality of B, if there is an injective function, but no bijective function, from A to B.

For example, the set N of all natural numbers has cardinality strictly less than its power set P(N), because g(n) = { n } is an injective function from N to P(N), and it can be shown that no function from N to P(N) can be bijective (see picture). By a similar argument, N has cardinality strictly less than the cardinality of the set R of all real numbers. For proofs, see Cantor's diagonal argument or Cantor's first uncountability proof.

If |A| ≤ |B| and |B| ≤ |A|, then |A| = |B| (a fact known as Schröder–Bernstein theorem). The axiom of choice is equivalent to the statement that |A| ≤ |B| or |B| ≤ |A| for every A, B.[11][12]

One of Cantor's most important results was that the cardinality of the continuum (${\displaystyle {\mathfrak {c}}}$) is greater than that of the natural numbers (${\displaystyle \aleph _{0}}$); that is, there are more real numbers R than natural numbers N. Namely, Cantor showed that ${\displaystyle {\mathfrak {c}}=2^{\aleph _{0}}=\beth _{1}}$ (see Beth one) satisfies:

${\displaystyle 2^{\aleph _{0}}>\aleph _{0}}$

(see Cantor's diagonal argument or Cantor's first uncountability proof).

The continuum hypothesis states that there is no cardinal number between the cardinality of the reals and the cardinality of the natural numbers, that is,

${\displaystyle 2^{\aleph _{0}}=\aleph _{1}}$

However, this hypothesis can neither be proved nor disproved within the widely accepted ZFC axiomatic set theory, if ZFC is consistent.

Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to the number of points in any segment of that line, but that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space. These results are highly counterintuitive, because they imply that there exist proper subsets and proper supersets of an infinite set S that have the same size as S, although S contains elements that do not belong to its subsets, and the supersets of S contain elements that are not included in it.

The first of these results is apparent by considering, for instance, the tangent function, which provides a one-to-one correspondence between the interval (−½π, ½π) and R (see also Hilbert's paradox of the Grand Hotel).

The second result was first demonstrated by Cantor in 1878, but it became more apparent in 1890, when Giuseppe Peano introduced the space-filling curves, curved lines that twist and turn enough to fill the whole of any square, or cube, or hypercube, or finite-dimensional space. These curves are not a direct proof that a line has the same number of points as a finite-dimensional space, but they can be used to obtain such a proof.

Cantor also showed that sets with cardinality strictly greater than ${\displaystyle {\mathfrak {c}}}$ exist (see his generalized diagonal argument and theorem). They include, for instance:

• the set of all subsets of R, i.e., the power set of R, written P(R) or 2^R
• the set R^R of all functions from R to R

Both have cardinality

${\displaystyle 2^{\mathfrak {c}}=\beth _{2}>{\mathfrak {c}}}$

(see Beth two).

The cardinal equalities ${\displaystyle {\mathfrak {c}}^{2}={\mathfrak {c}},}$ ${\displaystyle {\mathfrak {c}}^{\aleph _{0}}={\mathfrak {c}},}$ and ${\displaystyle {\mathfrak {c}}^{\mathfrak {c}}=2^{\mathfrak {c}}}$ can be demonstrated using cardinal arithmetic:

${\displaystyle {\mathfrak {c}}^{2}=\left(2^{\aleph _{0}}\right)^{2}=2^{2\times {\aleph _{0}}}=2^{\aleph _{0}}={\mathfrak {c}},}$

${\displaystyle {\mathfrak {c}}^{\aleph _{0}}=\left(2^{\aleph _{0}}\right)^{\aleph _{0}}=2^{{\aleph _{0}}\times {\aleph _{0}}}=2^{\aleph _{0}}={\mathfrak {c}},}$

${\displaystyle {\mathfrak {c}}^{\mathfrak {c}}=\left(2^{\aleph _{0}}\right)^{\mathfrak {c}}=2^{{\mathfrak {c}}\times \aleph _{0}}=2^{\mathfrak {c}}.}$