Typetoken distinctionFrom Wikipedia, the free encyclopediaContents1 Sequence 11.1 Examples and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.1 Important examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Indexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.3 Specifying a sequence by recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Formal denition and basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.2 Finite and innite. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.3 Increasing and decreasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.4 Bounded . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.5 Other types of sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Limits and convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3.1 Denition of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3.2 Applications and important results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3.3 Cauchy sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.5 Use in other elds of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.5.1 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.5.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.5.3 Linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.5.4 Abstract algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.5.5 Set theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.5.6 Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.5.7 Streams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.6 Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.7 Related concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.8 Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Symbol (formal) 142.1 Can words be modeled as formal symbols? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15iii CONTENTS2.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Typetoken distinction 163.1 Occurrences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Typography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 Well-formed formula 194.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.2 Propositional calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.3 Predicate logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.4 Atomic and open formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.5 Closed formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.6 Properties applicable to formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.7 Usage of the terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.12Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 244.12.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.12.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.12.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Chapter 1SequenceSequential redirects here. For the manual transmission, see Sequential manual transmission. For other uses, seeSequence (disambiguation).In mathematics, a sequence is an ordered collection of objects in which repetitions are allowed. Like a set, it containsmembers (also called elements, or terms). The number of elements (possibly innite) is called the length of thesequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at dierent positionsin the sequence. Formally, a sequence can be dened as a function whose domain is a countable totally ordered set,such as the natural numbers.For example, (M, A, R, Y) is a sequence of letters with the letter 'M' rst and 'Y' last.This sequence diers from(A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two dierent positions, is avalid sequence. Sequences can be nite, as in these examples, or innite, such as the sequence of all even positiveintegers (2, 4, 6,...). In computing and computer science, nite sequences are sometimes called strings, words or lists,the dierent names commonly corresponding to dierent ways to represent them into computer memory; innitesequences are also called streams. The empty sequence ( ) is included in most notions of sequence, but may beexcluded depending on the context.An innite sequence of real numbers (in blue). This sequence is neither increasing, nor decreasing, nor convergent, nor Cauchy. Itis, however, bounded.12 CHAPTER 1. SEQUENCE1.1 Examples and notationA sequence can be thought of as a list of elements with a particular order. Sequences are useful in a number ofmathematical disciplines for studying functions, spaces, and other mathematical structures using the convergenceproperties of sequences. In particular, sequences are the basis for series, which are important in dierential equationsand analysis. Sequences are also of interest in their own right and can be studied as patterns or puzzles, such as in thestudy of prime numbers.There are a number of ways to denote a sequence, some of which are more useful for specic types of sequences.One way to specify a sequence is to list the elements. For example, the rst four odd numbers form the sequence(1,3,5,7). This notation can be used for innite sequences as well. For instance, the innite sequence of positiveodd integers can be written (1,3,5,7,...). Listing is most useful for innite sequences with a pattern that can be easilydiscerned from the rst few elements. Other ways to denote a sequence are discussed after the examples.1.1.1 Important examples321 158A tiling with squares whose sides are successive Fibonacci numbers in length.There are many important integer sequences. The prime numbers are the natural numbers bigger than 1, that haveno divisors but 1 and themselves. Taking these in their natural order gives the sequence (2,3,5,7,11,13,17,...). Thestudy of prime numbers has important applications for mathematics and specically number theory.The Fibonacci numbers are the integer sequence whose elements are the sum of the previous two elements. The rsttwo elements are either 0 and 1 or 1 and 1 so that the sequence is (0,1,1,2,3,5,8,13,21,34,...).Other interesting sequences include the ban numbers, whose spellings do not contain a certain letter of the alphabet.For instance, the eban numbers (do not contain 'e') formthe sequence (2,4,6,30,32,34,36,40,42,...). Another sequencebased on the English spelling of the letters is the one based on their number of letters (3,3,5,4,4,3,5,5,4,3,6,6,8,...).For a list of important examples of integers sequences see On-line Encyclopedia of Integer Sequences.Other important examples of sequences include ones made up of rational numbers, real numbers, and complex num-bers. The sequence (.9,.99,.999,.9999,...) approaches the number 1. In fact, every real number can be written asthe limit of a sequence of rational numbers. It is this fact that allows us to write any real number as the limit of asequence of decimals. For instance, is the limit of the sequence (3,3.1,3.14,3.141,3.1415,...). The sequence for ,however, does not have any pattern that is easily discernible by eye, unlike the sequence (0.9,0.99,...).1.1. EXAMPLES AND NOTATION 31.1.2 IndexingOther notations can be useful for sequences whose pattern cannot be easily guessed, or for sequences that do not havea pattern such as the digits of . This section focuses on the notations used for sequences that are a map from a subsetof the natural numbers. For generalizations to other countable index sets see the following section and below.The terms of a sequence are commonly denoted by a single variable, say an, where the index n indicates the nthelement of the sequence.a1 element 1sta2 element 2nda3 element 3rd......an1 element (n-1)than element nthan+1 element (n+1)th......Indexing notation is used to refer to a sequence in the abstract. It is also a natural notation for sequences whoseelements are related to the index n (the elements position) in a simple way. For instance, the sequence of the rst 10square numbers could be written as(a1, a2, ..., a10), ak= k2.This represents the sequence (1,4,9,...100). This notation is often simplied further as(ak)10k=1, ak= k2.Here the subscript {k=1} and superscript 10 together tell us that the elements of this sequence are the ak such that k= 1, 2, ..., 10.Sequences can be indexed beginning and ending from any integer. The innity symbol is often used as the super-script to indicate the sequence including all integer k-values starting with a certain one. The sequence of all positivesquares is then denoted(ak)k=1, ak= k2.In cases where the set of indexing numbers is understood, such as in analysis, the subscripts and superscripts are oftenleft o. That is, one simply writes ak for an arbitrary sequence. In analysis, k would be understood to run from 1 to. However, sequences are often indexed starting from zero, as in(ak)k=0= (a0, a1, a2, ...).In some cases the elements of the sequence are related naturally to a sequence of integers whose pattern can be easilyinferred. In these cases the index set may be implied by a listing of the rst few abstract elements. For instance, thesequence of squares of odd numbers could be denoted in any of the following ways. (1, 9, 25, ...) (a1, a3, a5, ...), ak= k2 (a2k1)k=1, ak= k24 CHAPTER 1. SEQUENCE (ak)k=1, ak= (2k 1)2 ((2k 1)2)k=1Moreover, the subscripts and superscripts could have been left o in the third, fourth, and fth notations if theindexing set was understood to be the natural numbers.Finally, sequences can more generally be denoted by writing a set inclusion in the subscript, such as in(ak)kNThe set of values that the index can take on is called the index set. In general, the ordering of the elements ak isspecied by the order of the elements in the indexing set. When N is the index set, the element ak+1 comes after theelement ak since in N, the element (k+1) comes directly after the element k.1.1.3 Specifying a sequence by recursionSequences whose elements are related to the previous elements in a straightforward way are often specied usingrecursion. This is in contrast to the specication of sequence elements in terms of their position.To specify a sequence by recursion requires a rule to construct each consecutive element in terms of the ones beforeit. In addition, enough initial elements must be specied so that new elements of the sequence can be specied bythe rule. The principle of mathematical induction can be used to prove that a sequence is well-dened, which is tosay that that every element of the sequence is specied at least once and has a single, unambiguous value. Inductioncan also be used to prove properties about a sequence, especially for sequences whose most natural specication isby recursion.The Fibonacci sequence can be dened using a recursive rule along with two initial elements. The rule is that eachelement is the sum of the previous two elements, and the rst two elements are 0 and 1.an= an1 +an2 , with a0= 0 and a1= 1 .The rst ten terms of this sequence are 0,1,1,2,3,5,8,13,21, and 34. A more complicated example of a sequence thatis dened recursively is Recamans sequence, considered at the beginning of this section. We can dene Recamanssequence bya0= 0 and an= an1n if the result is positive and not already in the list. Otherwise, an= an1+n.Not all sequences can be specied by a rule in the form of an equation, recursive or not, and some can be quitecomplicated. For example, the sequence of prime numbers is the set of prime numbers in their natural order. Thisgives the sequence (2,3,5,7,11,13,17,...).One can also notice that the next element of a sequence is a function of the element before, and so we can write thenext element as :an+1= f(an)This functional notation can prove useful when one wants to prove the global monotony of the sequence.1.2 Formal denition and basic propertiesThere are many dierent notions of sequences in mathematics, some of which (e.g., exact sequence) are not coveredby the denitions and notations introduced below.1.2.1 Formal denitionA sequence is usually dened as a function whose domain is a countable totally ordered set, although in many disci-plines the domain is restricted, such as to the natural numbers. In real analysis a sequence is a function from a subset1.2. FORMAL DEFINITION AND BASIC PROPERTIES 5of the natural numbers to the real numbers.[1] In other words, a sequence is a map f(n) : N R. To recover ourearlier notation we might identify an = f(n) for all n or just write an : N R.In complex analysis, sequences are dened as maps from the natural numbers to the complex numbers (C).[2] Intopology, sequences are often dened as functions from a subset of the natural numbers to a topological space.[3]Sequences are an important concept for studying functions and, in topology, topological spaces. An important gener-alization of sequences, called a net, is to functions from a (possibly uncountable) directed set to a topological space.1.2.2 Finite and inniteThe length of a sequence is dened as the number of terms in the sequence.A sequence of a nite length n is also called an n-tuple. Finite sequences include the empty sequence ( ) that has noelements.Normally, the term innite sequence refers to a sequence which is innite in one direction, and nite in the otherthesequence has a rst element, but no nal element, it is called a singly innite, or one-sided (innite) sequence,when disambiguation is necessary. In contrast, a sequence that is innite in both directionsi.e. that has neither arst nor a nal elementis called a bi-innite sequence, two-way innite sequence, or doubly innite sequence.A function from the set Z of all integers into a set, such as for instance the sequence of all even integers ( , 4, 2,0, 2, 4, 6, 8 ), is bi-innite. This sequence could be denoted (2n)n= .One can interpret singly innite sequences as elements of the semigroup ring of the natural numbers R[N], and doublyinnite sequences as elements of the group ring of the integers R[Z]. This perspective is used in the Cauchy productof sequences.1.2.3 Increasing and decreasingA sequence is said to be monotonically increasing if each term is greater than or equal to the one before it. For asequence(an)n=1 this can be written as an an for all n N. If each consecutive term is strictly greater than(>) the previous term then the sequence is called strictly monotonically increasing. A sequence is monotonicallydecreasing if each consecutive term is less than or equal to the previous one, and strictly monotonically decreasingif each is strictly less than the previous. If a sequence is either increasing or decreasing it is called a monotonesequence. This is a special case of the more general notion of a monotonic function.The terms nondecreasing and nonincreasing are often used in place of increasing and decreasing in order to avoidany possible confusion with strictly increasing and strictly decreasing, respectively.1.2.4 BoundedIf the sequence of real numbers (an) is such that all the terms, after a certain one, are less than some real number M,then the sequence is said to be bounded from above. In less words, this means an M for all n greater than N forsome pair M and N. Any such M is called an upper bound.Likewise, if, for some real m, an m for all n greaterthan some N, then the sequence is bounded from below and any such m is called a lower bound.If a sequence isboth bounded from above and bounded from below then the sequence is said to be bounded.1.2.5 Other types of sequencesA subsequence of a given sequence is a sequence formed from the given sequence by deleting some of the elementswithout disturbing the relative positions of the remaining elements. For instance, the sequence of positive evenintegers (2,4,6,...) is a subsequence of the positive integers (1,2,3,...). The positions of some elements change whenother elements are deleted. However, the relative positions are preserved.Some other types of sequences that are easy to dene include:An integer sequence is a sequence whose terms are integers.A polynomial sequence is a sequence whose terms are polynomials.6 CHAPTER 1. SEQUENCEA positive integer sequence is sometimes called multiplicative if anm = an am for all pairs n,m such that n andm are coprime.[4] In other instances, sequences are often called multiplicative if an = na1 for all n. Moreover,the multiplicative Fibonacci sequence satises the recursion relation an = an an.1.3 Limits and convergenceMain article: Limit of a sequenceOne of the most important properties of a sequence is convergence. Informally, a sequence converges if it has a limit.5 10 15 20 250.00.20.40.60.81.0n + 12n2The plot of a convergent sequence (a) is shown in blue. Visually we can see that the sequence is converging to the limit zero as nincreases.Continuing informally, a (singly innite) sequence has a limit if it approaches some value L, called the limit, as nbecomes very large. That is, for an abstract sequence (an) (with n running from 1 to innity understood) the valueof an approaches L as n , denotedlimnan= L.More precisely, the sequence converges if there exists a limit L such that the remaining a's are arbitrarily close to Lfor some n large enough.If a sequence converges to some limit, then it is convergent; otherwise it is divergent.If an gets arbitrarily large as n we writelimnan= .1.3. LIMITS AND CONVERGENCE 7In this case we say that the sequence (an) diverges, or that it converges to innity.If an becomes arbitrarily small negative numbers (large in magnitude) as n we writelimnan= and say that the sequence diverges or converges to minus innity.1.3.1 Denition of convergenceFor sequences that can be written as (an)n=1 with an R we can write (an) with the indexing set understood as N.These sequences are most common in real analysis. The generalizations to other types of sequences are consideredin the following section and the main page Limit of a sequence.Let (an) be a sequence. In words, the sequence (an) is said to converge if there exists a number L such that no matterhow close we want the an to be to L (say -close where > 0), we can nd a natural number N such that all terms(aN+1, aN+2, ...) are further closer to L (within of L). [1] This is often written more compactly using symbols. Forinstance,for all > 0, there exists a natural number N such that L < an < L+ for all n N.In even more compact notation > 0, N N s.t. n N, |an L| < .The dierence in the denitions of convergence for (one-sided) sequences in complex analysis and metric spaces isthat the absolute value |an L| is interpreted as the distance in the complex plane ( zz ), and the distance underthe appropriate metric, respectively.1.3.2 Applications and important resultsImportant results for convergence and limits of (one-sided) sequences of real numbers include the following. Theseequalities are all true at least when both sides exist. For a discussion of when the existence of the limit on one sideimplies the existence of the other see a real analysis text such as can be found in the references.[1][5]The limit of a sequence is unique.limn(an bn) = limnan limnbnlimncan= c limnanlimn(anbn) = (limnan)(limnbn)limnanbn=limnanlimnbnprovided limnbn = 0limnapn= [limnan]pIf an bn for all n greater than some N, then limnan limnbn .(Squeeze Theorem) If an cn bn for all n > N, and limnan= limnbn= L, then limncn= L.If a sequence is bounded and monotonic then it is convergent.A sequence is convergent if and only if every subsequence is convergent.8 CHAPTER 1. SEQUENCEThe plot of a Cauchy sequence (X), shown in blue, as X versus n. Visually, we see that the sequence appears to be converging to thelimit zero as the terms in the sequence become closer together as n increases. In the real numbers every Cauchy sequence convergesto some limit.1.3.3 Cauchy sequencesMain article: Cauchy sequenceA Cauchy sequence is a sequence whose terms become arbitrarily close together as n gets very large. The notion ofa Cauchy sequence is important in the study of sequences in metric spaces, and, in particular, in real analysis. Oneparticularly important result in real analysis is Cauchy characterization of convergence for sequences:In the real numbers, a sequence is convergent if and only if it is Cauchy.In contrast, in the rational numbers, e.g. the sequence dened by x1 = 1 and xn = xn + 2/xn/2 is Cauchy, but has norational limit, cf. here.1.4 SeriesMain article: Series (mathematics)A series is, informally speaking, the sum of the terms of a sequence. That is, adding the rst N terms of a (one-sided)sequence forms the Nth term of another sequence, called a series. Thus the N series of the sequence (a) results inanother sequence (SN) given by:S1= a1S2= a1 +a2S3= a1 +a2+a3......SN= a1 +a2+a3 + ......We can also write the nth term of the series as1.5. USE IN OTHER FIELDS OF MATHEMATICS 9SN=Nn=1an.Then the concepts used to talk about sequences, such as convergence, carry over to series (the sequence of partialsums) and the properties can be characterized as properties of the underlying sequences (such as (an) in the lastexample). The limit, if it exists, of an innite series (the series created from an innite sequence) is written aslimNSN=n=1an.1.5 Use in other elds of mathematics1.5.1 TopologySequence play an important role in topology, especially in the study of metric spaces. For instance:A metric space is compact exactly when it is sequentially compact.Afunction froma metric space to another metric space is continuous exactly when it takes convergent sequencesto convergent sequences.A metric space is a connected space if, whenever the space is partitioned into two sets, one of the two setscontains a sequence converging to a point in the other set.A topological space is separable exactly when there is a dense sequence of points.Sequences can be generalized to nets or lters. These generalizations allow one to extend some of the above theoremsto spaces without metrics.Product topologyA product space of a sequence of topological spaces is the cartesian product of the spaces equipped with a naturaltopology called the product topology.More formally, given a sequence of spaces {Xi} , dene X such thatX:=iIXi,is the set of sequences {xi} where each xi is an element of Xi . Let the canonical projections be written as pi :X Xi. Then the product topology on X is dened to be the coarsest topology (i.e. the topology with the fewestopen sets) for which all the projections pi are continuous. The product topology is sometimes called the Tychonotopology.1.5.2 AnalysisIn analysis, when talking about sequences, one will generally consider sequences of the form(x1, x2, x3, . . . ) or (x0, x1, x2, . . . )which is to say, innite sequences of elements indexed by natural numbers.10 CHAPTER 1. SEQUENCEIt may be convenient to have the sequence start with an index dierent from1 or 0. For example, the sequence denedby xn = 1/log(n) would be dened only for n 2. When talking about such innite sequences, it is usually sucient(and does not change much for most considerations) to assume that the members of the sequence are dened at leastfor all indices large enough, that is, greater than some given N.The most elementary type of sequences are numerical ones, that is, sequences of real or complex numbers. This typecan be generalized to sequences of elements of some vector space. In analysis, the vector spaces considered are oftenfunction spaces. Even more generally, one can study sequences with elements in some topological space.Sequence spacesMain article: Sequence spaceAsequence space is a vector space whose elements are innite sequences of real or complex numbers. Equivalently, itis a function space whose elements are functions from the natural numbers to the eld K of real or complex numbers.The set of all such functions is naturally identied with the set of all possible innite sequences with elements in K,and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalarmultiplication. All sequence spaces are linear subspaces of this space. Sequence spaces are typically equipped witha norm, or at least the structure of a topological vector space.The most important sequences spaces in analysis are the pspaces, consisting of the p-power summable sequences,with the p-norm. These are special cases of Lpspaces for the counting measure on the set of natural numbers.Other important classes of sequences like convergent sequences or null sequences form sequence spaces, respectivelydenoted c and c0, with the sup norm. Any sequence space can also be equipped with the topology of pointwiseconvergence, under which it becomes a special kind of Frchet space called FK-space.1.5.3 Linear algebraSequences over a eld may also be viewed as vectors in a vector space. Specically, the set of F-valued sequences(where F is a eld) is a function space (in fact, a product space) of F-valued functions over the set of natural numbers.1.5.4 Abstract algebraAbstract algebra employs several types of sequences, including sequences of mathematical objects such as groups orrings.Free monoidMain article: Free monoidIf A is a set, the free monoid over A (denoted A*, also called Kleene star of A) is a monoid containing all the nitesequences (or strings) of zero or more elements of A, with the binary operation of concatenation. The free semigroupA+ is the subsemigroup of A* containing all elements except the empty sequence.Exact sequencesMain article: Exact sequenceIn the context of group theory, a sequenceG0f1 G1f2 G2f3 fn Gnof groups and group homomorphisms is called exact if the image (or range) of each homomorphism is equal to thekernel of the next:1.6. TYPES 11im(fk) = ker(fk+1)Note that the sequence of groups and homomorphisms may be either nite or innite.Asimilar denition can be made for certain other algebraic structures. For example, one could have an exact sequenceof vector spaces and linear maps, or of modules and module homomorphisms.Spectral sequencesMain article: Spectral sequenceIn homological algebra and algebraic topology, aspectralsequence is a means of computing homology groupsby taking successive approximations. Spectral sequences are a generalization of exact sequences, and since theirintroduction by Jean Leray (1946), they have become an important research tool, particularly in homotopy theory.1.5.5 Set theoryAn ordinal-indexed sequence is a generalization of a sequence. If is a limit ordinal and X is a set, an -indexedsequence of elements of X is a function from to X. In this terminology an -indexed sequence is an ordinarysequence.1.5.6 ComputingAutomata or nite state machines can typically be thought of as directed graphs, with edges labeled using some specicalphabet, . Most familiar types of automata transition from state to state by reading input letters from , followingedges with matching labels; the ordered input for such an automaton forms a sequence called a word (or input word).The sequence of states encountered by the automaton when processing a word is called a run. A nondeterministicautomaton may have unlabeled or duplicate out-edges for any state, giving more than one successor for some inputletter. This is typically thought of as producing multiple possible runs for a given word, each being a sequence ofsingle states, rather than producing a single run that is a sequence of sets of states; however, 'run' is occasionally usedto mean the latter.1.5.7 StreamsInnite sequences of digits (or characters) drawn from a nite alphabet are of particular interest in theoretical com-puter science. They are often referred to simply as sequences or streams, as opposed to nite strings. Innite binarysequences, for instance, are innite sequences of bits (characters drawn from the alphabet {0, 1}). The set C = {0,1} of all innite, binary sequences is sometimes called the Cantor space.An innite binary sequence can represent a formal language (a set of strings) by setting the n th bit of the sequence to1 if and only if the n th string (in shortlex order) is in the language. This representation is useful in the diagonalizationmethod for proofs.[6]1.6 Types1-sequenceArithmetic progressionCauchy sequenceFarey sequenceFibonacci sequence12 CHAPTER 1. SEQUENCEGeometric progressionLook-and-say sequenceThueMorse sequence1.7 Related conceptsList (computing)Ordinal-indexed sequenceRecursion (computer science)TupleSet theory1.8 OperationsCauchy productLimit of a sequence1.9 See alsoEnumerationNet (topology) (a generalization of sequences)On-Line Encyclopedia of Integer SequencesPermutationRecurrence relationSequence spaceSet (mathematics)1.10 References[1] Gaughan, Edward. 1.1 Sequences and Convergence. Introduction to Analysis. AMS (2009). ISBN 0-8218-4787-2.[2] Edward B. Sa & Arthur David Snider (2003). Chapter 2.1. Fundamentals of Complex Analysis. ISBN 01-390-7874-6.[3] James R. Munkres. Chapters 1&2. Topology. ISBN 01-318-1629-2.[4] Lando, Sergei K. 7.4 Multiplicative sequences. Lectures on generating functions. AMS. ISBN 0-8218-3481-9.[5] Dawikins, Paul. Series and Sequences. Pauls Online Math Notes/Calc II (notes). Retrieved 18 December 2012.[6] Oazer, Kemal. FORMAL LANGUAGES, AUTOMATAANDCOMPUTATION: DECIDABILITY (PDF). cmu.edu.Carnegie-Mellon University. Retrieved 24 April 2015.1.11. EXTERNAL LINKS 131.11 External linksThe dictionary denition of sequence at WiktionaryHazewinkel, Michiel, ed. (2001), Sequence, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4The On-Line Encyclopedia of Integer SequencesJournal of Integer Sequences (free)Sequence at PlanetMath.org.Chapter 2Symbol (formal)For other uses see Symbol (disambiguation)Symbols andstrings of symbolsWell-formed formulasTheoremsThis diagram shows the syntactic entities that may be constructed from formal languages. The symbols and strings of symbols maybe broadly divided into nonsense and well-formed formulas. A formal language can be thought of as identical to the set of itswell-formed formulas. The set of well-formed formulas may be broadly divided into theorems and non-theorems.A logical symbol is a fundamental concept in logic, tokens of which may be marks or a conguration of markswhich form a particular pattern. Although the term symbol in common use refers at some times to the idea beingsymbolized, and at other times to the marks on a piece of paper or chalkboard which are being used to express that142.1. CAN WORDS BE MODELED AS FORMAL SYMBOLS? 15idea; in the formal languages studied in mathematics and logic, the term symbol refers to the idea, and the marksare considered to be a token instance of the symbol. In logic, symbols build literal utility to illustrate ideas.Symbols of a formal language need not be symbols of anything. For instance there are logical constants which do notrefer to any idea, but rather serve as a form of punctuation in the language (e.g. parentheses). Symbols of a formallanguage must be capable of being specied without any reference to any interpretation of them.A symbol or string of symbols may comprise a well-formed formula if it is consistent with the formation rules of thelanguage.In a formal system a symbol may be used as a token in formal operations. The set of formal symbols in a formallanguage is referred to as an alphabet (hence each symbol may be referred to as a letter)[1]A formal symbol as used in rst-order logic may be a variable (member from a universe of discourse), a constant, afunction (mapping to another member of universe) or a predicate (mapping to T/F).Formal symbols are usually thought of as purely syntactic structures, composed into larger structures using a formalgrammar, though sometimes they may be associated with an interpretation or model (a formal semantics).2.1 Can words be modeled as formal symbols?The move to view units in natural language (e.g. English) as formal symbols was initiated by Noam Chomsky (itwas this work that resulted in the Chomsky hierarchy in formal languages). The generative grammar model lookedupon syntax as autonomous from semantics. Building on these models, the logician Richard Montague proposed thatsemantics could also be constructed on top of the formal structure:There is in my opinion no important theoretical dierence between natural languages and the articiallanguages of logicians; indeed, I consider it possible to comprehend the syntax and semantics of bothkinds of language within a single natural and mathematically precise theory. On this point I dier froma number of philosophers, but agree, I believe, with Chomsky and his associates. [2]This is the philosophical premise underlying Montague grammar.However, this attempt to equate linguistic symbols with formal symbols has been challenged widely, particularly inthe tradition of cognitive linguistics, by philosophers like Stevan Harnad, and linguists like George Lako and RonaldLangacker.2.2 References[1] John Hopcroft, Rajeev Motwani and Jerey Ullman, Introduction to Automata Theory, Languages, and Computation, 2000[2] Richard Montague, Universal Grammar, 19702.3 See alsoList of mathematical symbolsChapter 3Typetoken distinctionthumbtime=6In disciplines such as logic, linguistics, metalogic, typography, and computer programming, the typetoken distinc-tion is a distinction that separates a descriptive concept from objects that instantiate the concept, seen as particularinstances of it. For example, the sentence the bicycle is in the garage refers to a token of the type named bicycle,while the sentence the bicycle is becoming more popular refers to the type.This distinction in computer programming between classes and objects is similar, though in this context, class mayrefer to a set of objects (with class-level attribute or operations) rather than a description of an object in the set.The words type, concept, property, quality, feature and attribute are all used in describing things. Some verbs t someof these words better than others. E.g. You might say a rose bush is a plant that instantiates the type(s), or embodiesthe concept(s), or exhibits the properties, or possesses the qualities, features or attributes thorny, owering andbushy.The term property is used ambiguously to mean property type (height in feet) and/or property instance(1.74). The term concept is probably used more often for the property type (height in feet) than the property163.1. OCCURRENCES 17instance.Types like thorny are often understood ontologically as concepts. Types exist in descriptions of objects, but notas tangible physical objects. A type may have many tokens. However, types are not directly producible as tokensare. One can show someone a particular bicycle, but cannot show someone the type bicycle, as in "the bicycle ispopular. It is often presumed that tokens exist in space and time as concrete physical objects. But tokens of the typesthought, tennis match, government and act of kindness don't t this presumption.Clarity requires us to distinguish between abstract types and the tokens or things that embody or exemplify types.If we hear that two people "have the same car", we may conclude that they have the same type of car (e.g. the samemake and model), or the same particular token of the car (e.g. they share a single vehicle). The distinction is usefulin other ways, during discussion of language.3.1 OccurrencesThere is a related distinction very closely connected with the type-token distinction. This distinction is the distinctionbetween an object, or type of object, and an occurrence of it. In this sense, an occurrence is not necessarily atoken. Quine discovered this distinction. However, he only gave what he called an articial, but convenient andadequate denition as an occurrence of x in y is an initial segment of y ending in x".[1] Quines proposed denition,known as The Prex Proposal, has not received the attention it deserves, but at least one counter-proposal has beenformulated.[2]Considering the sentence: "A rose is a rose is a rose".We may equally correctly state that there are eight or threewords in the sentence. There are, in fact, three word types in the sentence: rose, is and a. There are eight wordtokens in a token copy of the line. The line itself is a type. There are not eight word types in the line. It contains (asstated) only the three word types, 'a,' 'is and 'rose,' each of which is unique. So what do we call what there are eightof? They are occurrences of words. There are three occurrences of the word type 'a,' two of 'is and three of 'rose'.The need to distinguish tokens of types from occurrences of types arises, not just in linguistics, but whenever typesof things have other types of things occurring in them.[3] Reection on the simple case of occurrences of numerals isoften helpful.3.2 TypographyIn typography, the typetoken distinction is used to determine the presence of a text printed by movable type:[4]The dening criteria which a typographic print has to fulll is that of the type identity of the variousletter forms which make up the printed text. In other words: each letter form which appears in the texthas to be shown as a particular instance (token) of one and the same type which contains a reverseimage of the printed letter.3.3 See alsoCharles Sanders Peirces typetoken distinctionMetalogicFormalism (philosophy)Is-aClass (philosophy)Type theoryType physicalism18 CHAPTER 3. TYPETOKEN DISTINCTION3.4 Notes[1] Quine, Quiddities[2] See the Wikipedia article "Occurrences of numerals"[3] Stanford Encyclopedia of Philosophy, Types and Tokens[4] Brekle, Herbert E.: Die Prfeninger Weiheinschrift von 1119. Eine palographisch-typographische Untersuchung, Scripto-rium Verlag fr Kultur und Wissenschaft, Regensburg 2005, ISBN 3-937527-06-0, p. 233.5 ReferencesBaggin J., and Fosl, P. (2003) The Philosophers Toolkit. Blackwell: 171-73. ISBN 978-0-631-22874-5.Peper F., Lee J., Adachi S.,Isokawa T. (2004) Token-Based Computing on Nanometer Scales, Proceeding ofthe ToBaCo 2004 Workshop on Token Based Computing, Vol.1 pp. 118.3.6 External linksThe Stanford Encyclopedia of Philosophy: "Types and Tokens" by Linda Wetzel.Chapter 4Well-formed formulaSymbols andstrings of symbolsWell-formed formulasTheoremsThis diagram shows the syntactic entities which may be constructed from formal languages. The symbols and strings of symbolsmay be broadly divided into nonsense and well-formed formulas. A formal language can be thought of as identical to the set of itswell-formed formulas. The set of well-formed formulas may be broadly divided into theorems and non-theorems.In mathematical logic, a well-formed formula, shortly w, often simply formula, is a word (i.e. a nite sequenceof symbols from a given alphabet) that is part of a formal language.[1] A formal language can be considered to beidentical to the set containing all and only its formulas.A formula is a syntactic formal object that can be given a semantic meaning by means of semantics.1920 CHAPTER 4. WELL-FORMED FORMULA4.1 IntroductionA key use of formulae is in propositional logic and predicate logics such as rst-order logic. In those contexts, aformula is a string of symbols for which it makes sense to ask is true?", once any free variables in have beeninstantiated. In formal logic, proofs can be represented by sequences of formulas with certain properties, and thenal formula in the sequence is what is proven.Although the term formula may be used for written marks (for instance, on a piece of paper or chalkboard), it ismore precisely understood as the sequence being expressed, with the marks being a token instance of formula. It isnot necessary for the existence of a formula that there be any actual tokens of it. A formal language may thus have aninnite number of formulas regardless whether each formula has a token instance. Moreover, a single formula mayhave more than one token instance, if it is written more than once.Formulas are quite often interpreted as propositions (as, for instance, in propositional logic). However formulas aresyntactic entities, and as such must be specied in a formal language without regard to any interpretation of them.An interpreted formula may be the name of something, an adjective, an adverb, a preposition, a phrase, a clause, animperative sentence, a string of sentences, a string of names, etc.. A formula may even turn out to be nonsense, if thesymbols of the language are specied so that it does. Furthermore, a formula need not be given any interpretation.4.2 Propositional calculusThe formulas of propositional calculus, also called propositional formulas,[2] are expressions such as (A (B C)). Their denition begins with the arbitrary choice of a set V of propositional variables. The alphabet consists of theletters in V along with the symbols for the propositional connectives and parentheses "(" and ")", all of which areassumed to not be in V. The formulas will be certain expressions (that is, strings of symbols) over this alphabet.The formulas are inductively dened as follows:Each propositional variable is, on its own, a formula.If is a formula, then is a formula.If and are formulas, and is any binary connective, then ( ) is a formula. Here could be (but is notlimited to) the usual operators , , , or .This denition can also be written as a formal grammar in BackusNaur form, provided the set of variables is nite: ::= p | q | r | s | t | u | ... (the arbitrary nite set of propositional variables) ::= | | ( ) | ( ) | ( ) |( )Using this grammar, the sequence of symbols(((p q) (r s)) ( q s))is a formula, because it is grammatically correct. The sequence of symbols((p q) (qq))p))is not a formula, because it does not conform to the grammar.A complex formula may be dicult to read, owing to, for example, the proliferation of parentheses. To alleviatethis last phenomenon, precedence rules (akin to the standard mathematical order of operations) are assumed amongthe operators, making some operators more binding than others. For example, assuming the precedence (from mostbinding to least binding) 1. 2. 3. 4. . Then the formula(((p q) (r s)) ( q s))4.3. PREDICATE LOGIC 21may be abbreviated asp q r s q sThis is, however, only a convention used to simplify the written representation of a formula. If the precedence wasassumed, for example, to be left-right associative, in following order: 1. 2. 3. 4. , then the same formulaabove (without parentheses) would be rewritten as(p (q r)) (s (( q) ( s)))4.3 Predicate logicThe denition of a formula in rst-order logic QS is relative to the signature of the theory at hand.This signaturespecies the constant symbols, relation symbols, and function symbols of the theory at hand, along with the arities ofthe function and relation symbols.The denition of a formula comes in several parts. First, the set of terms is dened recursively. Terms, informally,are expressions that represent objects from the domain of discourse.1. Any variable is a term.2. Any constant symbol from the signature is a term3. an expression of the form f(t1,...,tn), where f is an n-ary function symbol, and t1,...,tn are terms, is again aterm.The next step is to dene the atomic formulas.1. If t1 and t2 are terms then t1=t2 is an atomic formula2. If R is an n-ary relation symbol, and t1,...,tn are terms, then R(t1,...,tn) is an atomic formulaFinally, the set of formulas is dened to be the smallest set containing the set of atomic formulas such that thefollowing holds:1. is a formula when is a formula2. ( ) and ( ) are formulas when and are formulas;3.x is a formula when x is a variable and is a formula;4.xis a formula whenx is a variable andis a formula (alternatively, xcould be dened as an abbreviationfor x ).If a formula has no occurrences of x or x , for any variable x , then it is called quantier-free. An existentialformula is a formula starting with a sequence of existential quantication followed by a quantier-free formula.4.4 Atomic and open formulasMain article: Atomic formulaAn atomic formula is a formula that contains no logical connectives nor quantiers, or equivalently a formula that hasno strict subformulas. The precise form of atomic formulas depends on the formal system under consideration; forpropositional logic, for example, the atomic formulas are the propositional variables. For predicate logic, the atomsare predicate symbols together with their arguments, each argument being a term.According to some terminology, an open formula is formed by combining atomic formulas using only logical con-nectives, to the exclusion of quantiers.[3] This has not to be confused with a formula which is not closed.22 CHAPTER 4. WELL-FORMED FORMULA4.5 Closed formulasMain article: Sentence (mathematical logic)Aclosed formula, also ground formula or sentence, is a formula in which there are no free occurrences of any variable.If A is a formula of a rst-order language in which the variables v1, ..., vn have free occurrences, then A preceded byv1 ... vn is a closure of A.4.6 Properties applicable to formulasA formula A in a language Q is valid if it is true for every interpretation of Q .A formula A in a language Q is satisable if it is true for some interpretation of Q .A formulaA of the language of arithmetic is decidable if it represents a decidable set, i.e. if there is aneective method which, given a substitution of the free variables of A, says that either the resulting instance ofA is provable or its negation is.4.7 Usage of the terminologyIn earlier works on mathematical logic (e.g.by Church[4]), formulas referred to any strings of symbols and amongthese strings, well-formed formulas were the strings that followed the formation rules of (correct) formulas.Several authors simply say formula.[5][6][7][8] Modern usages (especially in the context of computer science withmathematical software such as model checkers, automated theorem provers, interactive theorem provers) tend toretain of the notion of formula only the algebraic concept and to leave the question of well-formedness, i.e.of theconcrete string representation of formulas (using this or that symbol for connectives and quantiers, using this or thatparenthesizing convention, using Polish or inx notation, etc.) as a mere notational problem.However, the expression well-formed formulas can still be found in various works,[9][10][11] these authors using thename well-formed formula without necessarily opposing it to the old sense of formula as arbitrary string of symbols sothat it is no longer common in mathematical logic to refer to arbitrary strings of symbols in the old sense of formulas.The expression well-formed formulas (WFF) also pervaded in popular culture. Indeed, WFF is part of an esotericpun used in the name of the academic game "WFF 'N PROOF: The Game of Modern Logic, by Layman Allen,[12]developed while he was at Yale Law School (he was later a professor at the University of Michigan). The suite ofgames is designed to teach the principles of symbolic logic to children (in Polish notation).[13] Its name is an echoof whienpoof, a nonsense word used as a cheer at Yale University made popular in The Whienpoof Song and TheWhienpoofs.[14]4.8 See alsoGround expression4.9 Notes[1] Formulas are a standard topic in introductory logic, and are covered by all introductory textbooks, including Enderton(2001), Gamut (1990), and Kleene (1967)[2] First-order logic and automated theorem proving, Melvin Fitting, Springer, 1996[3] Handbook of the history of logic, (Vol 5, Logic from Russell to Church), Tarskis logic by Keith Simmons, D. Gabbay andJ. Woods Eds, p568 .[4] Alonzo Church, [1996] (1944), Introduction to mathematical logic, page 494.10. REFERENCES 23[5] Hilbert, David; Ackermann, Wilhelm (1950) [1937], Principles of Mathematical Logic, New York: Chelsea[6] Hodges, Wilfrid (1997), A shorter model theory, Cambridge University Press, ISBN 978-0-521-58713-6[7] Barwise, Jon, ed. (1982), Handbook of Mathematical Logic, Studies in Logic and the Foundations of Mathematics, Am-sterdam: North-Holland, ISBN 978-0-444-86388-1[8] Cori, Rene; Lascar, Daniel (2000), Mathematical Logic: A Course with Exercises, Oxford University Press, ISBN 978-0-19-850048-3[9] Enderton, Herbert [2001] (1972), A mathematical introduction to logic (2nd ed.), Boston, MA: Academic Press, ISBN978-0-12-238452-3[10] R. L. Simpson (1999), Essentials of Symbolic Logic, page 12[11] Mendelson, Elliott [2010] (1964), An Introduction to Mathematical Logic (5th ed.), London: Chapman & Hall[12] Ehrenburg 2002[13] More technically, propositional logic using the Fitch-style calculus.[14] Allen (1965) acknowledges the pun.4.10 ReferencesAllen, Layman E. (1965), Toward Autotelic Learning of Mathematical Logic by the WFF 'NPROOF Games,Mathematical Learning: Report of a Conference Sponsored by the Committee on Intellective Processes Researchof the Social Science Research Council, Monographs of the Society for Research in Child Development 30 (1):2941Boolos, George; Burgess, John; Jerey, Richard (2002), Computability and Logic (4th ed.), Cambridge Uni-versity Press, ISBN 978-0-521-00758-0Ehrenberg, Rachel (Spring 2002). Hes Positively Logical. Michigan Today (University of Michigan). Re-trieved 2007-08-19.Enderton, Herbert (2001), A mathematical introduction to logic (2nd ed.), Boston, MA: Academic Press, ISBN978-0-12-238452-3Gamut, L.T.F. (1990), Logic, Language, and Meaning, Volume 1: Introduction to Logic, University Of ChicagoPress, ISBN 0-226-28085-3Hodges, Wilfrid (2001), Classical Logic I: First-Order Logic, in Goble, Lou, The Blackwell Guide to Philo-sophical Logic, Blackwell, ISBN 978-0-631-20692-7Hofstadter, Douglas (1980), Gdel, Escher, Bach: An Eternal Golden Braid, Penguin Books, ISBN 978-0-14-005579-5Kleene, Stephen Cole (2002) [1967], Mathematical logic, New York: Dover Publications, ISBN 978-0-486-42533-7, MR 1950307Rautenberg, Wolfgang (2010), A Concise Introduction to Mathematical Logic (3rd ed.), New York: SpringerScience+Business Media, doi:10.1007/978-1-4419-1221-3, ISBN 978-1-4419-1220-64.11 External linksWell-Formed Formula for First Order Predicate Logic - includes a short Java quiz.Well-Formed Formula at ProvenMathWFF N PROOF game site24 CHAPTER 4. WELL-FORMED FORMULA4.12 Text and image sources, contributors, and licenses4.12.1 Text Sequence Source:https://en.wikipedia.org/wiki/Sequence?oldid=673701777 Contributors:AxelBoldt, Mav, Zundark, Tarquin, XJaM,Toby Bartels, Imran, Camembert, Youandme, Lir, Patrick, Michael Hardy, Ihcoyc, Poor Yorick, Nikai, EdH, Charles Matthews, Dys-prosia, Greenrd, Hyacinth, Zero0000, Sabbut, Garo, Robbot, Lowellian, MathMartin, Stewartadcock, Henrygb, Bkell, Tosha, Centrx,Giftlite, BenFrantzDale, Lupin, Herbee, Horatio, Edcolins, Vadmium, Leonard Vertighel, Manuel Anastcio, Alexf, Fudo, Melikamp,Sam Hocevar, Tsemii, Ross bencina, Jiy, TedPavlic, Paul August, JoeSmack, Elwikipedista~enwiki, Syp, Pjrich, Shanes, Jonathan Drain,Nk, Obradovic Goran, Haham hanuka, Zaraki~enwiki, Merope, Jumbuck, Reubot, Jet57, Olegalexandrov, Ringbang, Djsasso, Total-cynic, Oleg Alexandrov, Hoziron, Linas, Madmardigan53, MFH, Isnow, Graham87, Dpv, Mendaliv, Salix alba, Figs, VKokielov, Log-gie, Rsenington, RexNL, Pexatus, Fresheneesz, Kri, Ryvr, Chobot, Lightsup55, Krishnavedala, Wavelength, Michael Slone, Grubber,Arthur Rubin, JahJah, Pred, Finell, KHenriksson, Gelingvistoj, Chris the speller, Bluebot, Nbarth, Mcaruso, Suicidalhamster, SundarBot,Dreadstar, Fagstein, Just plain Bill, Xionbox, Dreftymac, Gco, CRGreathouse, CBM, Gregbard, Cydebot, Xantharius, Epbr123, KClier,Saber Cherry, Rlupsa, Marek69, Urdutext, Icep, Ste4k, Mutt Lunker, JAnDbot, Asnac, Coolhandscot, Martinkunev, VoABot II, Avjoska,JamesBWatson, Brusegadi, Minimiscience, Stdazi, DerHexer, J.delanoy, Trusilver, Suenm~enwiki, Ncmvocalist, Belovedfreak, Policron,JingaJenga, VolkovBot, ABF, AlnoktaBOT, Philip Trueman, Digby Tantrum, JhsBot, Isis4563, Wolfrock, Xiong Yingfei, Newbyguesses,SieBot, Scarian, Yintan, Xelgen, Outs, Paolo.dL, OKBot, Pagen HD, Wahrmund, Classicalecon, Atif.t2, Crambo0349, ClueBot, Justin WSmith, Fyyer, SuperHamster, Excirial, Estirabot, Jotterbot, Thingg, Downgrader, Aj00200, XLinkBot, Stickee, Rror, WikHead, Brent-smith101, Addbot, Non-dropframe, Kongr43gpen, Matj Grabovsk, Legobot, Luckas-bot, Yobot, Eric-Wester, 4th-otaku, AnomieBOT,Jim1138, Law, Materialscientist, E2eamon, ArthurBot, Ayda D, Xqbot, Omnipaedista, RibotBOT, Charvest, Shadowjams, Thehelpful-bot, Dan6hell66, Constructive editor, Mark Renier, Tal physdancer, SixPurpleFish, Pinethicket, BRUTE, SkyMachine, PiRSquared17,Roy McCoy, RjwilmsiBot, Tzfyr, EmausBot, John of Reading, GoingBatty, Wikipelli, K6ka, Brent Perreault, Nellandmice, Bethnim,Ida Shaw, Alpha Quadrant, KuduIO, D.Lazard, SporkBot, Wayne Slam, Donner60, Chewings72, ClueBot NG, Satellizer, Widr, MerlI-wBot, Helpful Pixie Bot, HMSSolent, Curb Chain, Calabe1992, Brad7777, Minsbot, Praxiphenes, EuroCarGT, Ven Seyranyan., Jegyao,DavyRalph, Graphium, Jochen Burghardt, Brirush, Mark viking, LoMaPh, Immonster, EricsonWillians, Emlynlee, Buscus 3, JackHoang,BemusedObserver, Some1Redirects4You and Anonymous: 210 Symbol (formal)Source: https://en.wikipedia.org/wiki/Symbol_(formal)?oldid=630172650Contributors: Dominus, Markhurd, Hy-acinth, Timrollpickering, Mukerjee, Rich Farmbrough, Ruud Koot, MithrandirMage, Arthur Rubin, SmackBot, CBM, Gregbard, Cy-debot, PamD, Nick Number, Calaka, R'n'B, Good Olfactory, Addbot, AnomieBOT, FrescoBot, Tijfo098, Kejia, Jiri 1984, Masssly,HMSSolent, Saehry, Wamiq, Hbb 1988 and Anonymous: 6 TypetokendistinctionSource: https://en.wikipedia.org/wiki/Type%E2%80%93token_distinction?oldid=671289816 Contributors: Markhurd,Hyacinth, Giftlite, Demitsu, Kwamikagami, Koavf, Shaggyjacobs, Chris the speller, Dcamp314, Levineps, Gregbard, Gun Powder Ma,DGG, R'n'B, Maurice Carbonaro, Antony-22, LokiClock, Anonymous Dissident, Philogo, PaulTanenbaum, Davidknag, Fratrep, Classi-calecon, SchreiberBike, St485, Addbot, Zoorlat~enwiki, BOOLE1847, Yobot, Pcap, Yotaloop, AnomieBOT, Aerodinamicista, Loveless,Omnipaedista, Dylan Hunt, T of Locri, Tesseract2, Milkunderwood, Repep, GlaedrH, BattyBot, NABRASA, POLY1956, A.safardoostand Anonymous: 23 Well-formed formula Source: https://en.wikipedia.org/wiki/Well-formed_formula?oldid=660313819 Contributors: Edward, MichaelHardy, DavidWBrooks, Charles Matthews, Wik, Hyacinth, Onebyone, Josh Cherry, Tobias Bergemann, Giftlite, Andy, Jewarnica,Abdull, Mh, Dolda2000, Spayrard, Linas, GregorB, BD2412, Qwertyus, MithrandirMage, SpuriousQ, IanManka, Ihope127, Trovatore,Pawyilee, Ripper234, Arthur Rubin, Otto ter Haar, SmackBot, Pokipsy76, Mhss, Ioscius, Jgoulden, B7T, Midnighttonight, CRGreathouse,CBM, Gregbard, Cydebot, Julian Mendez, Forgot, Nick Number, Behco, Laymanal, CountingPine, Epsilon0, Dessources, Philogo,Paradoctor, DeathByNukes, Kumioko (renamed), Martarius, ClueBot, PixelBot, Hans Adler, Jonverve, Hugo Herbelin, LheaJLove, Ad-dbot, Yobot, Ptbotgourou, Notacupcakebaker, GrouchoBot, Tkuvho, Jonesey95, Dude1818, Gamewizard71, BillyPreset, GoingBatty,Bamyers99, Pyy999, Wcherowi, Helpful Pixie Bot, Laymanallen, CitationCleanerBot, Blakehill, Jochen Burghardt and Anonymous: 384.12.2 Images File:Cauchy_sequence_illustration.svg Source: https://upload.wikimedia.org/wikipedia/commons/6/62/Cauchy_sequence_illustration.svg License: CC0 Contributors: Own work Original artist: Krishnavedala File:Cauchy_sequence_illustration2.svg Source: https://upload.wikimedia.org/wikipedia/commons/7/7a/Cauchy_sequence_illustration2.svg License: Public domain Contributors: Based on File:Cauchy_sequence_illustration2.png by Oleg Alexandrov Original artist: Own work File:Converging_Sequence_example.svg Source: https://upload.wikimedia.org/wikipedia/commons/e/e4/Converging_Sequence_example.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: Melikamp File:Fibonacci_blocks.svg Source: https://upload.wikimedia.org/wikipedia/commons/b/bc/Fibonacci_blocks.svg License: Public do-main Contributors: Own work Original artist: ElectroKid ( ) File:Fibonacci_spiral_34.svg Source: 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Original artist: MithrandirMage File:Logic_portal.svg Source: https://upload.wikimedia.org/wikipedia/commons/7/7c/Logic_portal.svg License: CC BY-SA 3.0 Con-tributors: Own work Original artist: Watchduck (a.k.a. Tilman Piesk) File:Mergefrom.svg Source: https://upload.wikimedia.org/wikipedia/commons/0/0f/Mergefrom.svg License: Public domain Contribu-tors: ? Original artist: ?4.12. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 254.12.3 Content license Creative Commons Attribution-Share Alike 3.0