The Union of Two Countable Sets Is Again Countable Contradiction

"Countable" redirects here. For the linguistic concept, see Count noun. For the statistical concept, see Count data. For the company, see Countable (app).

Mathematical set up that can be enumerated

In mathematics, a prepare is countable if it has the aforementioned cardinality (the number of elements of the set) every bit some subset of the set of natural numbers N = {0, 1, 2, 3, ...}. Equivalently, a set S is countable if in that location exists an injective function f : S → N from S to N; it simply means that every element in S corresponds to a different chemical element in Due north.

A countable set is either a finite set or a countably space fix. Whether finite or space, the elements of a countable prepare tin can always be counted 1 at a time and — although the counting may never finish due to the space number of the elements to be counted — every element of the set is associated with a unique natural number.

Georg Cantor introduced the concept of countable sets, contrasting sets that are countable with those that are uncountable. Today, countable sets form the foundation of a branch of mathematics called discrete mathematics.

A note on terminology [edit]

Although the terms "countable" and "countably infinite" as divers here are quite mutual, the terminology is not universal.^[one] An alternative style uses countable to mean what is hither called countably infinite, and at most countable to mean what is here called countable.^{[2] [3]} To avoid ambiguity, 1 may limit oneself to the terms "at virtually countable" and "countably space", although with respect to concision this is the worst of both worlds.^{[ citation needed ]} The reader is advised to bank check the definition in utilise when encountering the term "countable" in the literature.

The terms enumerable ^[4] and denumerable ^{[5] [half-dozen]} may also be used, east.g. referring to countable and countably infinite respectively,^[vii] but as definitions vary the reader is one time again advised to check the definition in use.^[8]

Definition [edit]

The most curtailed definition is in terms of cardinality. A fix Due south is countable if its cardinality |Due south| is less than or equal to ${\displaystyle \aleph _{0}}$ (aleph-zilch), the cardinality of the set of natural numbers N. A set S is countably infinite if ${\displaystyle |S|=\aleph _{0}}$ . A set is uncountable if it is non countable, i.e. its cardinality is greater than ${\displaystyle \aleph _{0}}$ ; the reader is referred to Uncountable set for further discussion.^[9]

For every fix S, the following propositions are equivalent:

• S is countable.^[5]
• There exists an injective office from Southward to Due north.^{[10] [11]}
• Due south is empty or in that location exists a surjective function from N to Southward.^[11]
• There exists a bijective mapping between Southward and a subset of Due north.^[12]
• S is either finite or countably infinite.^[13]

Similarly, the following propositions are equivalent:

History [edit]

In 1874, in his first set theory article, Cantor proved that the fix of existent numbers is uncountable, thus showing that not all infinite sets are countable.^[17] In 1878, he used ane-to-ane correspondences to define and compare cardinalities.^[18] In 1883, he extended the natural numbers with his infinite ordinals, and used sets of ordinals to produce an infinity of sets having different infinite cardinalities.^[xix]

Introduction [edit]

A set is a collection of elements, and may exist described in many ways. One way is but to list all of its elements; for instance, the set consisting of the integers three, 4, and 5 may be denoted {3, 4, v}, chosen roster grade.^[20] This is only constructive for small sets, yet; for larger sets, this would be fourth dimension-consuming and mistake-prone. Instead of listing every single element, sometimes an ellipsis ("...") is used to correspond many elements between the starting element and the end element in a set, if the writer believes that the reader can hands guess what ... represents; for example, {1, 2, 3, ..., 100} presumably denotes the set of integers from 1 to 100. Even in this case, however, it is still possible to list all the elements, because number of elements in the set is finite.

Some sets are infinite; these sets have more than n elements where n is any integer that tin exist specified. (No matter how large the specified integer n is, such equally north = ix×10³² , infinite sets have more than north elements.) For instance, the gear up of natural numbers, denotable by {0, i, ii, iii, 4, five, ...},^[a] has infinitely many elements, and we cannot apply any natural number to give its size. Notwithstanding, information technology turns out that infinite sets do have a well-divers notion of size (or more properly, cardinality, the technical term for the number of elements in a set), and non all infinite sets have the same cardinality.

Bijective mapping from integer to fifty-fifty numbers

To sympathise what this means, nosotros first examine what it does not mean. For example, there are infinitely many odd integers, infinitely many even integers, and (hence) infinitely many integers overall. Notwithstanding, it turns out that the number of even integers, which is the same as the number of odd integers, is too the aforementioned as the number of integers overall. This is because nosotros tin adjust things such that, for every integer, there is a distinct even integer:

$${\displaystyle \ldots \,-\!2\!\rightarrow \!-\!4,\,-\!1\!\rightarrow \!-\!ii,\,0\!\rightarrow \!0,\,1\!\rightarrow \!2,\,2\!\rightarrow \!4\,\cdots }$$

or, more than generally, ${\displaystyle due north\rightarrow 2n}$ (see movie). What we have done here is adjust the integers and the even integers into a i-to-one correspondence (or bijection), which is a office that maps betwixt two sets such that each element of each set corresponds to a single chemical element in the other set.

However, not all infinite sets have the same cardinality. For example, Georg Cantor (who introduced this concept) demonstrated that the existent numbers cannot be put into one-to-1 correspondence with the natural numbers (not-negative integers), and therefore that the gear up of existent numbers has a greater cardinality than the set of natural numbers.

Formal overview [edit]

By definition, a fix S is countable if there exists an injective function f : S → N from S to the natural numbers Due north = {0, 1, ii, 3, ...}. Information technology simply means that every element in South has the correspondence to a different chemical element in N .

Information technology might seem natural to divide the sets into different classes: put all the sets containing one element together; all the sets containing ii elements together; ...; finally, put together all infinite sets and consider them as having the same size. This view is non tenable, however, nether the natural definition of size.

To elaborate this, nosotros demand the concept of a bijection. Although a "bijection" may seem a more advanced concept than a number, the usual development of mathematics in terms of prepare theory defines functions before numbers, every bit they are based on much simpler sets. This is where the concept of a bijection comes in: define the correspondence

a ↔ 1, b ↔ 2, c ↔ 3

Since every element of {a, b, c} is paired with precisely 1 chemical element of {1, 2, three}, and vice versa, this defines a bijection.

We at present generalize this situation; we define that two sets are of the same size, if and but if there is a bijection between them. For all finite sets, this gives united states the usual definition of "the same size".

As for the case of space sets, consider the sets A = {1, two, 3, ... }, the set of positive integers, and B = {2, 4, half dozen, ... }, the fix of even positive integers. We claim that, nether our definition, these sets have the same size, and that therefore B is countably infinite. Recall that to prove this, nosotros need to exhibit a bijection between them. This can be achieved using the consignment northward ↔ iin, and then that

1 ↔ ii, 2 ↔ four, iii ↔ half dozen, 4 ↔ 8, ....

As in the before instance, every element of A has been paired off with precisely one element of B, and vice versa. Hence they have the aforementioned size. This is an example of a set of the same size equally i of its proper subsets, which is incommunicable for finite sets.

As well, the set of all ordered pairs of natural numbers (the Cartesian product of ii sets of natural numbers, N × N) is countably space, as can be seen by following a path like the i in the picture:

The resulting mapping proceeds as follows:

0 ↔ (0, 0), 1 ↔ (one, 0), 2 ↔ (0, 1), 3 ↔ (2, 0), 4 ↔ (i, 1), 5 ↔ (0, 2), six ↔ (3, 0), ....

This mapping covers all such ordered pairs.

This form of triangular mapping recursively generalizes to n-tuples of natural numbers, i.e., (a ₁, a ₂, a _iii, ..., a _{due north}) where a_i and n are natural numbers, by repeatedly mapping the kickoff two elements of a n-tuple to a natural number. For example, (0, 2, 3) can be written every bit ((0, ii), 3). So (0, 2) maps to 5 and then ((0, 2), 3) maps to (5, iii), then (5, three) maps to 39. Since a unlike two-tuple, that is a pair such equally (a, b), maps to a different natural number, a difference betwixt two n-tuples by a single element is enough to ensure the northward-tuples being mapped to different natural numbers. So, an injection from the set of north-tuples to the fix of natural numbers N is proved. For the set of n-tuples made by the Cartesian production of finitely many different sets, each element in each tuple has the correspondence to a natural number, so every tuple can be written in natural numbers and then the same logic is applied to prove the theorem.

Theorem  —The Cartesian production of finitely many countable sets is countable.^{[21] [b]}

The set of all integers Z and the set of all rational numbers Q may intuitively seem much bigger than Northward. Just looks can exist deceiving. If a pair is treated equally the numerator and denominator of a vulgar fraction (a fraction in the form of a/b where a and b ≠ 0 are integers), then for every positive fraction, we tin can come up with a distinct natural number corresponding to it. This representation also includes the natural numbers, since every natural number is likewise a fraction Northward/1. And then nosotros can conclude that in that location are exactly as many positive rational numbers equally in that location are positive integers. This is too truthful for all rational numbers, equally tin can be seen below.

Theorem  — Z (the prepare of all integers) and Q (the set of all rational numbers) are countable.^[c]

In a similar manner, the set of algebraic numbers is countable.^{[23] [d]}

Sometimes more than 1 mapping is useful: a ready A to be shown equally countable is one-to-one mapped (injection) to another set B, then A is proved every bit countable if B is one-to-one mapped to the set of natural numbers. For case, the set of positive rational numbers can easily be one-to-one mapped to the set up of natural number pairs (ii-tuples) because p/q maps to (p, q). Since the set of natural number pairs is one-to-one mapped (actually one-to-one correspondence or bijection) to the set of natural numbers every bit shown in a higher place, the positive rational number gear up is proved as countable.

Theorem  —Any finite matrimony of countable sets is countable.^{[24] [25] [e]}

With the foresight of knowing that there are uncountable sets, nosotros can wonder whether or not this concluding result can be pushed any further. The answer is "yes" and "no", we can extend information technology, but we need to assume a new precept to do so.

Theorem  —(Bold the precept of countable choice) The union of countably many countable sets is countable.^[f]

For case, given countable sets a, b, c, ...

Enumeration for countable number of countable sets

Using a variant of the triangular enumeration nosotros saw higher up:

• a ₀ maps to 0
• a ₁ maps to 1
• b ₀ maps to 2
• a _ii maps to iii
• b _i maps to 4
• c ₀ maps to 5
• a ₃ maps to 6
• b ₂ maps to vii
• c _i maps to 8
• d ₀ maps to 9
• a ₄ maps to 10
• ...

This only works if the sets a, b, c, ... are disjoint. If not, and so the matrimony is fifty-fifty smaller and is therefore besides countable by a previous theorem.

We need the axiom of countable pick to index all the sets a, b, c, ... simultaneously.

Theorem  —The set up of all finite-length sequences of natural numbers is countable.

This set is the union of the length-one sequences, the length-2 sequences, the length-three sequences, each of which is a countable set (finite Cartesian product). Then we are talking nearly a countable wedlock of countable sets, which is countable by the previous theorem.

Theorem  —The ready of all finite subsets of the natural numbers is countable.

The elements of whatever finite subset can be ordered into a finite sequence. At that place are simply countably many finite sequences, and then also there are only countably many finite subsets.

Theorem  —Let S and T be sets.

1. If the part f : Southward → T is injective and T is countable then S is countable.
2. If the function m : S → T is surjective and South is countable then T is countable.

These follow from the definitions of countable set up as injective / surjective functions.^[1000]

Cantor's theorem asserts that if A is a set and P(A) is its power set, i.e. the set of all subsets of A, then there is no surjective office from A to P(A). A proof is given in the article Cantor's theorem. Equally an immediate consequence of this and the Basic Theorem higher up we have:

Proposition  —The set P(N) is not countable; i.e. it is uncountable.

For an elaboration of this effect see Cantor's diagonal argument.

The gear up of real numbers is uncountable,^[h] and so is the set of all infinite sequences of natural numbers.

Minimal model of fix theory is countable [edit]

If in that location is a prepare that is a standard model (see inner model) of ZFC set theory, then there is a minimal standard model (encounter Constructible universe). The Löwenheim–Skolem theorem tin be used to testify that this minimal model is countable. The fact that the notion of "uncountability" makes sense even in this model, and in particular that this model K contains elements that are:

• subsets of M, hence countable,
• merely uncountable from the point of view of M,

was seen as paradoxical in the early days of ready theory, see Skolem's paradox for more.

The minimal standard model includes all the algebraic numbers and all effectively computable transcendental numbers, likewise as many other kinds of numbers.

Full orders [edit]

Countable sets can be totally ordered in various ways, for example:

• Well-orders (meet also ordinal number):
  • The usual order of natural numbers (0, 1, ii, iii, 4, 5, ...)
  • The integers in the gild (0, ane, two, 3, ...; −1, −2, −3, ...)
• Other (not well orders):
  • The usual lodge of integers (..., −3, −2, −1, 0, ane, two, 3, ...)
  • The usual guild of rational numbers (Cannot exist explicitly written every bit an ordered list!)

In both examples of well orders hither, any subset has a least element; and in both examples of non-well orders, some subsets do not have a least element. This is the key definition that determines whether a total society is also a well order.

See also [edit]

• Aleph number
• Counting
• Hilbert's paradox of the 1000 Hotel
• Uncountable fix

Notes [edit]

1. ^ Since there is an obvious bijection between N and N* = {one, 2, three, ...}, it makes no divergence whether one considers 0 a natural number or non. In any case, this commodity follows ISO 31-11 and the standard convention in mathematical logic, which takes 0 as a natural number.
2. ^ Proof: Discover that North × N is countable as a upshot of the definition because the office f : N × Northward → N given by f(yard, due north) = ii ^g 3 ^north is injective.^[22] It then follows that the Cartesian product of any ii countable sets is countable, because if A and B are 2 countable sets at that place are surjections f : N → A and g : N → B . So

   f × g : N × N → A × B

   is a surjection from the countable prepare North × N to the ready A × B and the Corollary implies A × B is countable. This event generalizes to the Cartesian product of whatever finite collection of countable sets and the proof follows by consecration on the number of sets in the collection.

3. ^ Proof: The integers Z are countable because the function f : Z → N given by f(n) = ii ⁿ if n is not-negative and f(n) = 3^−northward if northward is negative, is an injective role. The rational numbers Q are countable because the function thou : Z × N → Q given by g(m, n) = g/(n + 1) is a surjection from the countable set Z × North to the rationals Q .
4. ^ Proof: Per definition, every algebraic number (including complex numbers) is a root of a polynomial with integer coefficients. Given an algebraic number ${\displaystyle \alpha }$ , let ${\displaystyle a_{0}ten^{0}+a_{ane}x^{1}+a_{2}ten^{2}+\cdots +a_{n}x^{north}}$ be a polynomial with integer coefficients such that ${\displaystyle \blastoff }$ is the gth root of the polynomial, where the roots are sorted by absolute value from small to large, then sorted by argument from minor to large. We can define an injection (i. e. 1-to-one) function f : A → Q given by ${\displaystyle f(\alpha )=2^{thousand-1}\cdot 3^{a_{0}}\cdot v^{a_{one}}\cdot 7^{a_{2}}\cdots {p_{due north+two}}^{a_{n}}}$ , while ${\displaystyle p_{n}}$ is the due north-th prime.
5. ^ Proof: If A_i is a countable set for each i in I={one,...,n}, and so for each northward there is a surjective function one thousand_i  : Northward → A_i and hence the office

   $${\displaystyle G:I\times \mathbf {Northward} \to \bigcup _{i\in I}A_{i},}$$

   

   given by G(i, m) = yard_i (m) is a surjection. Since I × N is countable, the union ${\textstyle \bigcup _{i\in I}A_{i}}$ is countable.
6. ^ Proof: As in the finite case, but I=North and we employ the axiom of countable selection to pick for each i in N a surjection g_i from the non-empty collection of surjections from N to A_i .
7. ^ Proof: For (1) observe that if T is countable there is an injective function h : T → N. And then if f : S → T is injective the limerick h o f : South → Northward is injective, so S is countable. For (2) observe that if S is countable, either Southward is empty or there is a surjective function h : N → Southward. And so if thousand : S → T is surjective, either Due south and T are both empty, or the composition yard o h : N → T is surjective. In either case T is countable.
8. ^ See Cantor'southward first uncountability proof, and also Finite intersection holding#Applications for a topological proof.

Citations [edit]

1. ^ Manetti, Marco (19 June 2015). Topology. Springer. p. 26. ISBN978-three-319-16958-3.
2. ^ Rudin 1976, Chapter 2
3. ^ Tao & 2016 181 harvnb error: no target: CITEREFTao2016181 (help)
4. ^ Kamke 1950, p. 2
5. ^ ^{a b} Lang 1993, §ii of Affiliate I
6. ^ Apostol 1969, p. 23, Chapter 1.14
7. ^ Thierry, Vialar (4 April 2017). Handbook of Mathematics. BoD - Books on Need. p. 24. ISBN978-2-9551990-ane-v.
8. ^ Mukherjee, Subir Kumar (2009). First Course in Real Assay. Academic Publishers. p. 22. ISBN978-81-89781-ninety-three.
9. ^ Yaqub, Aladdin M. (24 October 2014). An Introduction to Metalogic. Broadview Press. ISBN978-ane-4604-0244-3.
10. ^ Singh, Tej Bahadur (17 May 2019). Introduction to Topology. Springer. p. 422. ISBN978-981-thirteen-6954-4.
11. ^ ^{a b} Katzourakis, Nikolaos; Varvaruca, Eugen (2 Jan 2018). An Illustrative Introduction to Modern Analysis. CRC Press. ISBN978-1-351-76532-nine.
12. ^ Halmos 1960, p. 91
13. ^ Weisstein, Eric Due west. "Countable Set". mathworld.wolfram.com . Retrieved 2020-09-06 .
14. ^ Kamke 1950, p. 2
15. ^ Dlab, Vlastimil; Williams, Kenneth S. (9 June 2020). Invitation To Algebra: A Resource Compendium For Teachers, Advanced Undergraduate Students And Graduate Students In Mathematics. World Scientific. p. 8. ISBN978-981-12-1999-3.
16. ^ Tao & 2016 182 harvnb mistake: no target: CITEREFTao2016182 (assist)
17. ^ Stillwell, John C. (2010), Roads to Infinity: The Mathematics of Truth and Proof, CRC Press, p. 10, ISBN9781439865507, Cantor's discovery of uncountable sets in 1874 was 1 of the most unexpected events in the history of mathematics. Before 1874, infinity was non fifty-fifty considered a legitimate mathematical subject field past most people, and then the need to distinguish between countable and uncountable infinities could not have been imagined.
18. ^ Cantor 1878, p. 242.
19. ^ Ferreirós 2007, pp. 268, 272–273.
20. ^ "What Are Sets and Roster Class?". expii. 2021-05-09. Archived from the original on 2020-09-18.
21. ^ Halmos 1960, p. 92
22. ^ Avelsgaard 1990, p. 182
23. ^ Kamke 1950, pp. 3–4
24. ^ Avelsgaard 1990, p. 180
25. ^ Fletcher & Patty 1988, p. 187

References [edit]

• Apostol, Tom K. (June 1969), Multi-Variable Calculus and Linear Algebra with Applications , Calculus, vol. 2 (2d ed.), New York: John Wiley + Sons, ISBN978-0-471-00007-v
• Avelsgaard, Carol (1990), Foundations for Advanced Mathematics, Scott, Foresman and Company, ISBN0-673-38152-8
• Cantor, Georg (1878), "Ein Beitrag zur Mannigfaltigkeitslehre", Journal für die Reine und Angewandte Mathematik, 1878 (84): 242–248, doi:x.1515/crelle-1878-18788413
• Ferreirós, José (2007), Labyrinth of Thought: A History of Ready Theory and Its Role in Mathematical Thought (second revised ed.), Birkhäuser, ISBN978-3-7643-8349-7
• Fletcher, Peter; Patty, C. Wayne (1988), Foundations of Higher Mathematics, Boston: PWS-KENT Publishing Visitor, ISBN0-87150-164-3
• Halmos, Paul R. (1960), Naive Set up Theory, D. Van Nostrand Visitor, Inc Reprinted past Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition). Reprinted by Martino Fine Books, 2011. ISBN 978-1-61427-131-4 (Paperback edition).
• Kamke, Erich (1950), Theory of Sets, Dover serial in mathematics and physics, New York: Dover, ISBN978-0486601410
• Lang, Serge (1993), Existent and Functional Analysis, Berlin, New York: Springer-Verlag, ISBN0-387-94001-4
• Rudin, Walter (1976), Principles of Mathematical Analysis, New York: McGraw-Hill, ISBN0-07-054235-10
• Tao, Terence (2016). "Space sets". Analysis I (Tertiary ed.). Singapore: Springer. pp. 181–210. ISBN978-981-x-1789-6.

Expect upwardly countable in Wiktionary, the free dictionary.

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Source: https://en.wikipedia.org/wiki/Countable_set

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