The set of all Platonic solids has 5 elements. Thus the cardinality of is 5 or, in symbols, .
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set contains 3 elements, and therefore has a cardinality of 3. Beginning in the late 19th century, this concept was generalized to infinite sets, which allows one to distinguish between different types of infinity, and to perform arithmetic on them. There are two approaches to cardinality: one which compares sets directly using bijections and injections, and another which uses cardinal numbers.[1]
The cardinality of a set may also be called its size, when no confusion with other notions of size[2] is possible.
The cardinality of a set is usually denoted , with a vertical bar on each side;[3] this is the same notation as absolute value, and the meaning depends on context. The cardinality of a set may alternatively be denoted by , , , or .
^Weisstein, Eric W. "Cardinal Number". MathWorld.
^Such as length and area in geometry. – A line of finite length is a set of points that has infinite cardinality.
^"Cardinality | Brilliant Math & Science Wiki". brilliant.org. Retrieved 2020-08-23.
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...
rank among the infinite cardinals. Cardinality is defined in terms of bijective functions. Two sets have the same cardinality if, and only if, there is...
aleph (ℵ). The cardinality of the natural numbers is ℵ0 (read aleph-nought or aleph-zero or aleph-null), the next larger cardinality of a well-ordered...
In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers R {\displaystyle \mathbb {R} } , sometimes called...
Calculating the exact cardinality of the distinct elements of a multiset requires an amount of memory proportional to the cardinality, which is impractical...
numbers is the same size (cardinality) as the set of integers: they are both countable sets. Cantor gave two proofs that the cardinality of the set of integers...
uncountable cardinal κ: κ is weakly compact. for every λ<κ, natural number n ≥ 2, and function f: [κ]n → λ, there is a set of cardinality κ that is homogeneous...
for β < α. Its cardinality is the limit of the cardinalities of these number classes. If n is finite, the n-th number class has cardinality ℵ n − 1 {\displaystyle...
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numbers to X. The cardinality of X is neither finite nor equal to ℵ 0 {\displaystyle \aleph _{0}} (aleph-null). The set X has cardinality strictly greater...
of subsets of S of cardinality less than or equal to κ is sometimes denoted by Pκ(S) or [S]κ, and the set of subsets with cardinality strictly less than...
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of cardinality κ {\displaystyle \kappa } . Then X {\displaystyle X} has cardinality at most 2 2 κ {\displaystyle 2^{2^{\kappa }}} and cardinality at most...
are no intermediate cardinal numbers between ℵ 0 {\displaystyle \aleph _{0}} and the cardinality of the continuum (the cardinality of the set of real numbers):...
Equinumerous sets are said to have the same cardinality (number of elements). The study of cardinality is often called equinumerosity (equalness-of-number)...
particular maximum-cardinality matching in which prioritized vertices are matched first. The problem of finding a maximum-cardinality matching in hypergraphs...
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In linguistics, and more precisely in traditional grammar, a cardinal numeral (or cardinal number word) is a part of speech used to count. Examples in...
{\displaystyle \aleph _{0}} (the cardinality of the set of natural numbers), and the cardinality of the continuum, that is, the cardinality of the set R {\displaystyle...
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