Inmathematics, thecardinality of aset is the number of its elements. The cardinality of a set may also be called itssize, when no confusion with other notions of size is possible.[a] Beginning in the late 19th century, this concept of size was generalized toinfinite sets, allowing one to distinguish between different types of infinity and to performarithmetic on them. Nowadays, infinite sets are encountered in almost all parts of mathematics, even those that may seem to be unrelated. Familiar examples are provided by mostnumber systems andalgebraic structures (natural numbers,rational numbers,real numbers,vector spaces, etc.), as well as, in geometry, bylines,line segments andcurves, which are considered as the sets of their points.
There are two approaches to describing cardinality: one which usescardinal numbers and another which compares sets directly using functions between them, eitherbijections orinjections.The former states the size as a number; the latter compares their relative size and led to the discovery of different sizes of infinity.[1] For example, the sets and are the same size as they each contain 3elements (the first approach) and there is a bijection between them (the second approach).
In English, the termcardinality originates from thepost-classical Latincardinalis, meaning “principal” or “chief,” which derives fromcardo, a noun meaning “hinge.” In Latin,cardo referred to something central or pivotal, both literally and metaphorically. This concept of centrality passed intomedieval Latin and then into English, wherecardinal came to describe things considered to be, in some sense, fundamental, such ascardinal virtues,cardinal directions, and (in the grammatical sense)cardinal numbers.[2] The last of which referred to numbers used for counting (e.g., one, two, three),[3] as opposed toordinal numbers, which express order (e.g., first, second, third),[4] andnominal numbers used for labeling (without meaning).
In mathematics, the notion of cardinality was first introduced byGeorg Cantor in the late 19th century, wherein he used the used the termMächtigkeit, which may be translated as “magnitude” or “power", though Cantor credited the term to a work byJakob Steiner onprojective geometry.[5][6][7] The termscardinality andcardinal number were eventually adopted from the grammatical sense, and later translations would use these terms.
A crude sense of cardinality, an awareness that groups of things or events compare with other groups by containing more, fewer, or the same number of instances, is observed in a variety of present-day animal species, suggesting an origin millions of years ago.[8] Human expression of cardinality is seen as early as40000 years ago, with equating the size of a group with a group of recorded notches, or a representative collection of other things, such as sticks and shells.[9] The abstraction of cardinality as a number is evident by 3000 BCE, in Sumerianmathematics and the manipulation of numbers without reference to a specific group of things or events.[10]
From the 6th century BCE, the writings of Greek philosophers show hints of the cardinality of infinite sets. While they considered the notion of infinity as an endless series of actions, such as adding 1 to a number repeatedly, they did not consider the size of an infinite set of numbers to be a thing.[11] The ancient Greek notion of infinity also considered the division of things into parts repeated without limit. In Euclid'sElements,commensurability was described as the ability to compare the length of two line segments,a andb, as a ratio, as long as there were a third segment, no matter how small, that could be laid end-to-end a whole number of times into botha andb. But with the discovery ofirrational numbers, it was seen that even the infinite set of all rational numbers was not enough to describe the length of every possible line segment.[12]
One of the earliest explicit uses of a one-to-one correspondence is recorded inAristotle'sMechanics, known asAristotle's wheel paradox. The paradox can be briefly described as follows: A wheel is depicted as twoconcentric circles. The larger, outer circle is tangent to a horizontal line (e.g. a road that it rolls on), while the smaller, inner circle is rigidly affixed to the larger. Assuming the larger circle rolls along the line without slipping (or skidding) for one full revolution, the distances moved by both circles are the same: thecircumference of the larger circle. Further, the lines traced by the bottom-most point of each is the same length.[13] Since the smaller wheel does not skip any points, and no point on the smaller wheel is used more than once, there is a one-to-one correspondence between the two circles.
Galileo Galilei presented what was later coinedGalileo's paradox in his bookTwo New Sciences (1638), where he attempts to show that infinite quantities cannot be called greater or less than one another. He presents the paradox roughly as follows: asquare number is one which is the product of another number with itself, such as 4 and 9, which are the squares of 2 and 3 respectively. Then thesquare root of a square number is that multiplicand. He then notes that there are as many square numbers as there are square roots, since every square has its own root and every root its own square, while no square has more than one root and no root more than one square. But there are as many square roots as there are numbers, since every number is the square root of some square. He, however, concluded that this meant we could not compare the sizes of infinite sets, missing the opportunity to discover cardinality.[14]
Bernard Bolzano'sParadoxes of the Infinite (Paradoxien des Unendlichen, 1851) is generally considered the first rigorous introduction of sets to mathematics. In his work, he (among other things) expanded on Galileo's paradox, and introduced one-to-one correspondence of infinite sets, for example between theintervals and by the relation. However, he resisted saying these sets wereequinumerous, and his work is generally considered to have been uninfluential in mathematics of his time.[15][16]
To better understand infinite sets, a notion of cardinality was formulatedc. 1880 byGeorg Cantor, the originator ofset theory. He examined the process of equating two sets with abijection, a one-to-one correspondence between the elements of two sets. In 1891, with the publication ofhis diagonal argument, he demonstrated that there are sets of numbers that cannot be placed in one-to-one correspondence with the set of natural numbers, i.e., there are "uncountable sets" that contain more elements than there are in the infinite set of natural numbers.[17]
The cardinality, orcardinal number, of a set is generally denoted by, with avertical bar on each side.[18] (This is the same notation as forabsolute value; the meaning depends on context.) The notation means that the two sets and have the same cardinality. The cardinal number of a set may also be denoted by,,,, etc.It is conventional to recognize three kinds of cardinality:
While the cardinality of a finite set is simply its number of elements, extending that notion to infinite sets usually starts with defining comparison of sizes of arbitrary sets (some of which are possibly infinite).
Two sets have the same cardinality if there exists a one-to-one correspondence between the elements of and those (that is, abijection from to).[19] Such sets are said to beequipotent,equipollent, orequinumerous.
For example, the set of non-negativeeven numbers has the same cardinality as the set ofnatural numbers, since the function is a bijection from to (see picture).
For finite sets and, ifsome bijection exists from to, theneach injective or surjective function from to is a bijection. This is no longer true for infinite and. For example, the function from to, defined by is injective, but not surjective since 2, for instance, is not mapped to, and from to, defined by (see:modulo operation) is surjective, but not injective, since 0 and 1 for instance both map to 0. Neither nor can challenge, which was established by the existence of.
has cardinality less than or equal to the cardinality of, if there exists an injective function from into.
If and, then (a fact known as theSchröder–Bernstein theorem). Theaxiom of choice is equivalent to the statement that or for every and.[20][21]
has cardinality strictly less than the cardinality of, if there is an injective function, but no bijective function, from to.
For example, the set of allnatural numbers has cardinality strictly less than itspower set, because is an injective function from to, and it can be shown that no function from to can be bijective (see picture). By a similar argument, has cardinality strictly less than the cardinality of the set of allreal numbers. For proofs, seeCantor's diagonal argument orCantor's first uncountability proof.
In the above section, "cardinality" of a set was defined functionally. In other words, it was not defined as a specific object itself. However, such an object can be defined as follows.
The relation of having the same cardinality is calledequinumerosity, and this is anequivalence relation on theclass of all sets. Theequivalence class of a setA under this relation, then, consists of all those sets which have the same cardinality asA. There are two ways to define the "cardinality of a set":
Assuming theaxiom of choice, the cardinalities of theinfinite sets are denoted
For eachordinal, is the least cardinal number greater than.
The cardinality of thenatural numbers is denotedaleph-null (), while the cardinality of thereal numbers is denoted by "" (a lowercasefraktur script "c"), and is also referred to as thecardinality of the continuum. Cantor showed, using thediagonal argument, that. We can show that, this also being the cardinality of the set of all subsets of the natural numbers.
Thecontinuum hypothesis says that, i.e. is the smallest cardinal number bigger than, i.e. there is no set whose cardinality is strictly between that of the integers and that of the real numbers. The continuum hypothesis isindependent ofZFC, a standard axiomatization of set theory; that is, it is impossible to prove the continuum hypothesis or its negation from ZFC—provided that ZFC is consistent. For more detail, see§ Cardinality of the continuum below.[22][23][24]
Our intuition gained fromfinite sets breaks down when dealing withinfinite sets. In the late 19th centuryGeorg Cantor,Gottlob Frege,Richard Dedekind and others rejected the view that the whole cannot be the same size as the part.[25][citation needed] One example of this isHilbert's paradox of the Grand Hotel.Indeed, Dedekind defined an infinite set as one that can be placed into a one-to-one correspondence with a strict subset (that is, having the same size in Cantor's sense); this notion of infinity is calledDedekind infinite. Cantor introduced the cardinal numbers, and showed—according to his bijection-based definition of size—that some infinite sets are greater than others. The smallest infinite cardinality is that of the natural numbers ().
One of Cantor's most important results was that thecardinality of the continuum () is greater than that of the natural numbers (); that is, there are more real numbersR than natural numbersN. Namely, Cantor showed that (seeBeth one) satisfies:
Thecontinuum hypothesis states that there is nocardinal number between the cardinality of the reals and the cardinality of the natural numbers, that is,
However, this hypothesis can neither be proved nor disproved within the widely acceptedZFCaxiomatic set theory, if ZFC is consistent.
Cardinal arithmetic can be used to show not only that the number of points in areal number line is equal to the number of points in anysegment 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 existproper subsets andproper supersets of an infinite setS that have the same size asS, althoughS contains elements that do not belong to its subsets, and the supersets ofS contain elements that are not included in it.
The first of these results is apparent by considering, for instance, thetangent function, which provides aone-to-one correspondence between theinterval (−½π, ½π) andR (see alsoHilbert's paradox of the Grand Hotel).
The second result was first demonstrated by Cantor in 1878, but it became more apparent in 1890, whenGiuseppe Peano introduced thespace-filling curves, curved lines that twist and turn enough to fill the whole of any square, or cube, orhypercube, 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 obtainsuch a proof.
Cantor also showed that sets with cardinality strictly greater than exist (see hisgeneralized diagonal argument andtheorem). They include, for instance:
Both have cardinality
Thecardinal equalities and can be demonstrated usingcardinal arithmetic:
IfA andB aredisjoint sets, then
From this, one can show that in general, the cardinalities ofunions andintersections are related by the following equation:[26]
Here denote a class of all sets, and denotes the class of all ordinal numbers.
We use the intersection of a class which is defined by, therefore.In this case
This definition allows also obtain a cardinality of any proper class, in particular
This definition is natural since it agrees with the axiom of limitation of size which implies bijection between and any proper class.
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