Cardinality

For other uses, see Cardinality (disambiguation). Cardinality_sentence_0

Comparing sets Cardinality_section_0

While the cardinality of a finite set is just the number of its elements, extending the notion to infinite sets usually starts with defining the notion of comparison of arbitrary sets (some of which are possibly infinite). Cardinality_sentence_1

Definition 1: |A| = |B| Cardinality_section_1

Cardinality_description_list_0

• 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, 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.Cardinality_item_0_0

Cardinality_description_list_1

• 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).Cardinality_item_1_1

Definition 2: |A| ≤ |B| Cardinality_section_2

Cardinality_description_list_2

• A has cardinality less than or equal to the cardinality of B, if there exists an injective function from A into B.Cardinality_item_2_2

Definition 3: |A| < |B| Cardinality_section_3

Cardinality_description_list_3

• A has cardinality strictly less than the cardinality of B, if there is an injective function, but no bijective function, from A to B.Cardinality_item_3_3

Cardinality_description_list_4

If |A| ≤ |B| and |B| ≤ |A|, then |A| = |B| (a fact known as Schröder–Bernstein theorem). Cardinality_sentence_2

The axiom of choice is equivalent to the statement that |A| ≤ |B| or |B| ≤ |A| for every A, B. Cardinality_sentence_3

Cardinal numbers Cardinality_section_4

Main article: Cardinal number Cardinality_sentence_4

In the above section, "cardinality" of a set was defined functionally. Cardinality_sentence_5

In other words, it was not defined as a specific object itself. Cardinality_sentence_6

However, such an object can be defined as follows. Cardinality_sentence_7

The relation of having the same cardinality is called equinumerosity, and this is an equivalence relation on the class of all sets. Cardinality_sentence_8

The equivalence class of a set A under this relation, then, consists of all those sets which have the same cardinality as A. Cardinality_sentence_9

There are two ways to define the "cardinality of a set": Cardinality_sentence_10

Cardinality_ordered_list_5

1. The cardinality of a set A is defined as its equivalence class under equinumerosity.Cardinality_item_5_5
2. A representative set is designated for each equivalence class. The most common choice is the initial ordinal in that class. This is usually taken as the definition of cardinal number in axiomatic set theory.Cardinality_item_5_6

Assuming the axiom of choice, the cardinalities of the infinite sets are denoted Cardinality_sentence_11

Finite, countable and uncountable sets Cardinality_section_5

If the axiom of choice holds, the law of trichotomy holds for cardinality. Cardinality_sentence_12

Thus we can make the following definitions: Cardinality_sentence_13

Infinite sets Cardinality_section_6

Cardinality of the continuum Cardinality_section_7

Main article: Cardinality of the continuum Cardinality_sentence_14

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, Cardinality_sentence_15

However, this hypothesis can neither be proved nor disproved within the widely accepted ZFC axiomatic set theory, if ZFC is consistent. Cardinality_sentence_16

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. Cardinality_sentence_17

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. Cardinality_sentence_18

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). Cardinality_sentence_19

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. Cardinality_sentence_20

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. Cardinality_sentence_21

Cardinality_description_list_6

• Cardinality_item_6_7
• the set of all subsets of R, i.e., the power set of R, written P(R) or 2Cardinality_item_6_8
• the set R of all functions from R to RCardinality_item_6_9

Both have cardinality Cardinality_sentence_22

Union and intersection Cardinality_section_9

If A and B are disjoint sets, then Cardinality_sentence_23

From this, one can show that in general, the cardinalities of unions and intersections are related by the following equation: Cardinality_sentence_24