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mathematical induction
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Mathematical induction can be informally illustrated by reference to the sequential effect of falling dominoes.Matt DeVos, Mathematical Induction, Simon Fraser UniversityGerardo con Diaz, Mathematical Induction, Harvard University
Mathematical induction is a mathematical proof technique. It is essentially used to prove that a property P(n) holds for every natural number n, i.e. for n = 0, 1, 2, 3, and so on. Metaphors can be informally used to understand the concept of mathematical induction, such as the metaphor of falling dominoes or climbing a ladder:The method of induction requires two cases to be proved. The first case, called the base case (or, sometimes, the basis), proves that the property holds for the number 0. The second case, called the induction step, proves that, if the property holds for one natural number n, then it holds for the next natural number n + 1. These two steps establish the property P(n) for every natural number n = 0, 1, 2, 3, ... The base step need not begin with zero. Often it begins with the number one, and it can begin with any natural number, establishing the truth of the property for all natural numbers greater than or equal to the starting number.The method can be extended to prove statements about more general well-founded structures, such as trees; this generalization, known as structural induction, is used in mathematical logic and computer science. Mathematical induction in this extended sense is closely related to recursion. Mathematical induction, in some form, is the foundation of all correctness proofs for computer programs.BOOK
- Dominoeffect.png -
Mathematical induction can be informally illustrated by reference to the sequential effect of falling dominoes.Matt DeVos, Mathematical Induction, Simon Fraser UniversityGerardo con Diaz, Mathematical Induction, Harvard University
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Although its name may suggest otherwise, mathematical induction should not be misconstrued as a form of inductive reasoning as used in philosophy (also see Problem of induction). Mathematical induction is an inference rule used in formal proofs. Proofs by mathematical induction are, in fact, examples of deductive reasoning.WEB, Suber, Peter, Mathematical Induction,weblink Earlham College, 26 March 2011, , Robert B.
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History
In 370 BC, Plato's Parmenides may have contained an early example of an implicit inductive proof.JOURNAL, Acerbi, Fabio, Plato: Parmenides 149a7-c3. A Proof by Complete Induction?,weblink Archive for History of Exact Sciences, 55, 2000, 57â€“76, The earliest implicit traces of mathematical induction may be found in Euclid'sWEB,weblink Euclid's Proof of the Infinitude of Primes (c. 300 BC), Chris K. Caldwell, utm.edu, 2016-02-28, proof that the number of primes is infinite and in Bhaskara's "cyclic method".{{harvp|Cajori|1918|p=197|ps=: 'The process of reasoning called "Mathematical Induction" has had several independent origins. It has been traced back to the Swiss Jakob (James) Bernoulli, the Frenchman B. Pascal and P. Fermat, and the Italian F. Maurolycus. [...] By reading a little between the lines one can find traces of mathematical induction still earlier, in the writings of the Hindus and the Greeks, as, for instance, in the "cyclic method" of Bhaskara, and in Euclid's proof that the number of primes is infinite.'}} An opposite iterated technique, counting down rather than up, is found in the Sorites paradox, where it was argued that if 1,000,000 grains of sand formed a heap, and removing one grain from a heap left it a heap, then a single grain of sand (or even no grains) forms a heap.An implicit proof by mathematical induction for arithmetic sequences was introduced in the al-Fakhri written by al-Karaji around 1000 AD, who used it to prove the binomial theorem and properties of Pascal's triangle.{{citation|title=The Development of Arabic Mathematics: Between Arithmetic and Algebra|volume=156|series=Boston Studies in the Philosophy of Science|publisher=Kluwer Academic Publishers|first=R.|last=Rashed|year=1994|isbn=9780792325659|pages=62â€“84|url=https://books.google.com/books?id=vSkClSvU_9AC&pg=PA62|contribution=Mathematical induction: al-KarajÄ« and al-SamawÊ¾al}}Mathematical Knowledge and the Interplay of Practices "The earliest implicit proof by mathematical induction was given around 1000 in a work by the Persian mathematician Al-Karaji"None of these ancient mathematicians, however, explicitly stated the induction hypothesis. Another similar case (contrary to what Vacca has written, as Freudenthal carefully showed) was that of Francesco Maurolico in his Arithmeticorum libri duo (1575), who used the technique to prove that the sum of the first n odd integers is n2. The first explicit formulation of the principle of induction was given by Pascal in his TraitÃ© du triangle arithmÃ©tique (1665). Another Frenchman, Fermat, made ample use of a related principle, indirect proof by infinite descent. The induction hypothesis was also employed by the Swiss Jakob Bernoulli, and from then on it became more or less well known. The modern rigorous and systematic treatment of the principle came only in the 19th century, with George Boole,"It is sometimes required to prove a theorem which shall be true whenever a certain quantity n which it involves shall be an integer or whole number and the method of proof is usually of the following kind. 1st. The theorem is proved to be true when n = 1. 2ndly. It is proved that if the theorem is true when n is a given whole number, it will be true if n is the next greater integer. Hence the theorem is true universally. . .. This species of argument may be termed a continued sorites" (Boole circa 1849 Elementary Treatise on Logic not mathematical pages 40â€“41 reprinted in Grattan-Guinness, Ivor and Bornet, GÃ©rard (1997), George Boole: Selected Manuscripts on Logic and its Philosophy, BirkhÃ¤user Verlag, Berlin, {{isbn|3-7643-5456-9}}) Augustus de Morgan, Charles Sanders Peirce,NEWS, Peirce, C. S., Charles Sanders Peirce, On the Logic of Number,weblink American Journal of Mathematics, 4, 1881< m}}" or assuming that a set of m elements has an element.Complete induction is equivalent to ordinary mathematical induction as described above, in the sense that a proof by one method can be transformed into a proof by the other. Suppose there is a proof of P(n) by complete induction. Let Q(n) mean "P(m) holds for all m such that {{nowrap|0 â‰¤ m â‰¤ n}}". Then Q(n) holds for all n if and only if P(n) holds for all n, and our proof of P(n) is easily transformed into a proof of Q(n) by (ordinary) induction. If, on the other hand, P(n) had been proven by ordinary induction, the proof would already effectively be one by complete induction: P(0) is proved in the base case, using no assumptions, and {{nowrap|P(n + 1)}} is proved in the inductive step, in which one may assume all earlier cases but need only use the case P(n).Example: Fibonacci numbers
Complete induction is most useful when several instances of the inductive hypothesis are required for each inductive step. For example, complete induction can be used to show that
F_n = frac{varphi^n - psi^n}{varphi - psi}
where Fn is the nth Fibonacci number, {{nowrap|1=Ï† = (1 + âˆš{{overline|5}})/2}} (the golden ratio) and {{nowrap|1=Ïˆ = (1 âˆ’ âˆš{{overline|5}})/2}} are the roots of the polynomial {{nowrap|x2 âˆ’ x âˆ’ 1}}. By using the fact that {{nowrap|1=F'n+2 = F'n+1 + F'n}} for each {{nowrap|n âˆˆ N}}, the identity above can be verified by direct calculation for {{nowrap|F'n+2}} if one assumes that it already holds for both F'n+1 and F'n. To complete the proof, the identity must be verified in the two base cases {{nowrap|1=n = 0}} and {{nowrap|1=n = 1}}.Example: prime factorization
Another proof by complete induction uses the hypothesis that the statement holds for all smaller n more thoroughly. Consider the statement that "every natural number greater than 1 is a product of (one or more) prime numbers", which is the "existence" part of the fundamental theorem of arithmetic. For proving the inductive step, the induction hypothesis is that for a given {{nowrap|n > 1}} the statement holds for all smaller {{nowrap|n > 1}}. If m is prime then it is certainly a product of primes, and if not, then by definition it is a product: {{nowrap|1=m = n1n2}}, where neither of the factors is equal to 1; hence neither is equal to m, and so both are smaller than m. The induction hypothesis now applies to n1 and n2, so each one is a product of primes. Thus m is a product of products of primes; therefore itself a product of primes.Example: dollar amounts revisited
We shall look to prove the same example as (#Example: forming dollar amounts by coins|above), this time with a variant called strong induction. The statement remains the same:S(n): ,,ngeq12 to ,exists, a,binmathbb{N}. ,, n=4a+5bHowever, there will be slight differences with the structure and assumptions of the proof. Let us begin with the base case.Base case: Show that S(k) holds for k=12,13,14,15.
begin{align}
4 cdot 3+5 cdot 0=124 cdot 2+5 cdot 1=134 cdot 1+5 cdot 2=144 cdot 0+5 cdot 3=15end{align}The base case holds.Induction hypothesis: Given some j>15 such that S(m) holds for all m with 12leq m< j.Inductive step: Prove that S(j) holds.Choosing m=j-4, and observing that 15< j to 12leq j-4 1. However, the logic of the inductive step is incorrect for n = 1, because the statement that "the two sets overlap" is false (there are only n + 1 = 2 horses prior to either removal, and after removal the sets of one horse each do not overlap).Formalization
In second-order logic, we can write down the "axiom of induction" as follows:
displaystyleforall PBigl( P(0) land forall k bigl( P(k) to P(k+1)bigr ) to forall n bigl(P(n)bigr)Bigr),
where P(.) is a variable for predicates involving one natural number and k and n are variables for natural numbers.In words, the base case P(0) and the inductive step (namely, that the induction hypothesis P(k) implies P(k + 1)) together imply that P(n) for any natural number n. The axiom of induction asserts that the validity of inferring that P(n) holds for any natural number n from the base case and the inductive step.Note that the first quantifier in the axiom ranges over predicates rather than over individual numbers. This is a second-order quantifier, which means that this axiom is stated in second-order logic. Axiomatizing arithmetic induction in first-order logic requires an axiom schema containing a separate axiom for each possible predicate. The article Peano axioms contains further discussion of this issue.The axiom of structural induction for the natural numbers was first formulated by Peano, who used it to specify the natural numbers together with four other axioms saying that (1) 0 is a natural number, (2) the successor function s of every natural number yields a natural number (s(x)=x+1), (3) the successor function is injective, and (4) 0 is not in the range of s.In first-order ZFC set theory, quantification over predicates is not allowed, but we can still phrase induction by quantification over sets:
forall A Bigl( 0 in A land forall k in mathbb{N} bigl( k in A to (k+1) in A bigr) to mathbb{N}subseteq ABigr)
A may be read as a set representing a proposition, and containing natural numbers, for which the proposition holds. This is not an axiom, but a theorem, given that natural numbers are defined in the language of ZFC set theory by axioms, analogous to Peano's.Transfinite induction
The principle of complete induction is not only valid for statements about natural numbers, but for statements about elements of any well-founded set, that is, a set with an irreflexive relation < that contains no infinite descending chains. Any set of cardinal numbers is well-founded, which includes the set of natural numbers.Applied to a well-founded set, it can be formulated as a single step:- Show that if some statement holds for all {{nowrap|m < n}}, then the same statement also holds for n.
- when n is a minimal element, i.e. there is no element smaller than n;
- when n has a direct predecessor, i.e. the set of elements which are smaller than n has a largest element;
- when n has no direct predecessor, i.e. n is a so-called limit ordinal.
Equivalence with the well-ordering principle{{anchor|Proof of mathematical induction}}
The principle of mathematical induction is usually stated as an axiom of the natural numbers; see Peano axioms. However, it can be proved from the well-ordering principle. Indeed, suppose the following:- The set of natural numbers is well-ordered.
- Every natural number is either 0, or {{nowrap|n + 1}} for some natural {{nowrap|number n}}.
- For any natural number n, {{nowrap|n + 1}} is greater {{nowrap|than n}}.
- P(0) holds and
- whenever P(m) is true then {{nowrap|P(m + 1}}) is also true,
See also
Notes
{{Reflist}}References
Introduction
- BOOK, J., Franklin, James Franklin (philosopher), 2011, Proof in Mathematics: An Introduction,weblink Kew Books, Sydney, 978-0-646-54509-7, A. Daoud, (Ch. 8.)
- {{springer|title=Mathematical induction|id=p/m062640}}
- BOOK, Hans, Hermes, Introduction to Mathematical Logic, London, Springer, Hochschultext, 1431-4657, 978-3540058199, 1973, harv,
- BOOK, Donald E., Knuth, Donald Knuth, 1997, The Art of Computer Programming, Volume 1: Fundamental Algorithms, 3rd, Addison-Wesley, 978-0-201-89683-1, (Section 1.2.1: Mathematical Induction, pp. 11â€“21.)
- BOOK, Andrey N., Kolmogorov, Andrey Kolmogorov, Silverman, R. A. (trans., ed.), 1975, Introductory Real Analysis, Dover, New York, 978-0-486-61226-3, Sergei V. Fomin, (Section 3.8: Transfinite induction, pp. 28â€“29.)
History
- JOURNAL, Archive for History of Exact Sciences, 55, 2000, 57â€“76, Plato: Parmenides 149a7-c3. A Proof by Complete Induction?, F., Acerbi, 10.1007/s004070000020,
- JOURNAL, The Origin of Mathematical Induction, W. H., 2974308, Bussey, The American Mathematical Monthly, 24, 5, 1917, 199â€“207, 10.2307/2974308,
- JOURNAL, Florian, Cajori, Florian Cajori, Origin of the Name "Mathematical Induction", 2972638, The American Mathematical Monthly, 1918, 25, 5, 197â€“201, 10.2307/2972638, harv,
- JOURNAL, Fowler D., 1994, Could the Greeks Have Used Mathematical Induction? Did They Use It?, Physis, XXXI, 253â€“265,
- JOURNAL, Hans, Freudenthal, Hans Freudenthal, 1953, Zur Geschichte der vollstÃ¤ndigen Induction, Archives Internationales d'Histoire des Sciences, 6, 17â€“37,
- Katz, Victor J. (1998). History of Mathematics: An Introduction. Addison-Wesley. {{isbn|0-321-01618-1}}.
- NEWS, Peirce, C. S., Charles Sanders Peirce, On the Logic of Number,weblink American Journal of Mathematics, 4, 1881, 1â€“4, 85â€“95, 10.2307/2369151, 1507856, 2369151, Reprinted (CP 3.252-88), (W 4:299-309).
- JOURNAL, Nachum L., Rabinovitch, Rabbi Levi Ben Gershon and the origins of mathematical induction, Archive for History of Exact Sciences, 1970, 6, 3, 237â€“248, 10.1007/BF00327237,
- JOURNAL, Roshdi, Rashed, L'induction mathÃ©matique: al-KarajÄ«, as-Samaw'al, Archive for History of Exact Sciences, 1972, 9, 1, 1â€“21, 10.1007/BF00348537, French,
- BOOK, Paul, Shields, 1997, Peirce's Axiomatization of Arithmetic, Houser, Studies in the Logic of Charles S. Peirce, harv, etal,
- JOURNAL, Unguru, S., 1991, Greek Mathematics and Mathematical Induction, Physis, XXVIII, 273â€“289,
- JOURNAL, Unguru, S., 1994, Fowling after Induction, Physis, XXXI, 267â€“272,
- JOURNAL, G., Vacca, Maurolycus, the First Discoverer of the Principle of Mathematical Induction, Bulletin of the American Mathematical Society, 1909, 16, 70â€“73, 10.1090/S0002-9904-1909-01860-9, 2,
- JOURNAL, The Use of Mathematical Induction by AbÅ« KÄmil ShujÄ' Ibn Aslam (850-930), Mohammad, Yadegari, Isis, 69, 2, 1978, 259â€“262, 10.1086/352009, 230435,
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pages=85â€“95 | mr=1507856 | Shields|1997}} Giuseppe Peano, and Richard Dedekind.DescriptionThe simplest and most common form of mathematical induction infers that a statement involving a natural number n holds for all values of n. The proof consists of two steps:
ExampleMathematical induction can be used to prove that the following statement, P(n), holds for all natural numbers n.
0 + 1 + 2 + cdots + n = frac{n(n + 1)}{2}.
P(n) gives a formula for the sum of the natural numbers less than or equal to number n. The proof that P(n) is true for each natural number n proceeds as follows.Base case: Show that the statement holds for n = 0 (taking 0 as a natural).P(0) is easily seen to be true:
0 = frac{0cdot(0 + 1)}{2},.
Inductive step: Show that if P(k) holds, then also {{nowrap|P(k + 1)}} holds. This can be done as follows.Assume P(k) holds (for some unspecified value of k). It must then be shown that {{nowrap|P(k + 1)}} holds, that is:
(0 + 1 + 2 + cdots + k )+ (k+1) = frac{(k+1)((k+1) + 1)}{2}.
Using the induction hypothesis that P(k) holds, the left-hand side can be rewritten to:
frac{k(k + 1)}{2} + (k+1),.
Algebraically:
Variants{{no footnotes|section|date=July 2013}}In practice, proofs by induction are often structured differently, depending on the exact nature of the property to be proven.Induction basis other than 0 or 1If one wishes to prove a statement not for all natural numbers but only for all numbers n greater than or equal to a certain number b, then the proof by induction consists of:
Example: forming dollar amounts by coinsAssume an infinite supply of 4- and 5-dollar coins. Induction can be used to prove that any whole amount of dollars greater than 12 can be formed by a combination of such coins. The amount n is chosen to begin on 12 as the statement does not hold true for every lower number; in particular, it is violated for n=11.In more precise terms, we wish to show that for any amount ngeq12 there exist natural numbers a,b such that n=4a+5b, where 0 is included as a natural number. The statement to be shown is thus:
S(n): ,, ngeq12 to exists, a,binmathbb{N}.,, n=4a+5b
Base case: Showing that S(k) holds for k=12 is trivial: let a=3 and b=0. Then, 4 cdot 3+5 cdot 0=12.Step case: Given that S(k) holds for some value of k geq 12 (induction hypothesis), prove that S(k+1) holds, too. That is, given that k=4a+5b for some natural numbers a,b, prove that there exist natural numbers a_1,b_1 such that k+1=4a_1+5b_1.Here we need to consider two cases.For the first case, assume that ageq 1. By some algebraic manipulation and by assumption, we see that in that case
begin{align}
k&=4a+5bk+1&=4a+5b+1&=4a+5b-4+5&=4(a-1)+5(b+1)&=4a_1+5b_1end{align}where a_1=a-1 and b_1=b+1 are natural numbers.This shows that to add 1 to the total amountâ€”any amount whatsoever, so long as it is greater than 12â€”it is sufficient to remove a single 4-dollar coin while adding a 5-dollar coin. However, this construction fails in the case that a=0, or in words, when there is no 4-dollar coin. So it remains to prove the case a=0. Then k=5b geq 12, which implies that b geq 3.
begin{align}
k&=5bk+1&=5b+1&=5b-15+16&=5(b-3)+4cdot4&=4cdot4+5(b-3)&=4a_1+5b_1end{align}where a_1=4 and b_1=b-3 are again natural numbers.The above calculation shows that in the case there are no 4-dollar coins, we can add 1 to the amount by removing three 5-dollar coins while adding four 4-dollar coins.Thus, with the inductive step, we have shown that S(k) implies S(k+1) for all natural numbers k geq 12, and the proof is complete. Q.E.D.This proof cannot be modified to replace the minimum amount of 12 dollar to any lower value m:for m=11, the base case is not provable, since there is no way to get 11 dollar by 4- and 5-dollar coins only;for m=10, the second case in the induction step will not work any longer, since k=5b geq 10 will not imply b geq 3;let alone for even lower m.Induction on more than one counterIt is sometimes desirable to prove a statement involving two natural numbers, n and m, by iterating the induction process. That is, one proves a base case and an inductive step for n, and in each of those proves a base case and an inductive step for m. See, for example, the proof of commutativity accompanying addition of natural numbers. More complicated arguments involving three or more counters are also possible.Infinite descentThe method of infinite descent is a variation of mathematical induction which was used by Pierre de Fermat. It is used to show that some statement Q(n) is false for all natural numbers n. Its traditional form consists of showing that if Q(n) is true for some natural number n, it also holds for some strictly smaller natural number m. Because there are no infinite decreasing sequences of natural numbers, this situation would be impossible, showing by contradiction that Q(n) cannot be true for any n. The validity of this method can be verified from the usual principle of mathematical induction. Using mathematical induction on the statement P(n) defined as "Q(m) is false for all natural numbers m less than or equal to n", it follows that P(n) holds for all n, which means that Q(n) is false for every natural number n.Prefix inductionThe most common form of proof by mathematical induction requires proving in the inductive step that
forall k (P(k) to P(k+1))
whereupon the induction principle "automates" n applications of this step in getting from P(0) to P(n). This could be called "predecessor induction" because each step proves something about a number from something about that number's predecessor.A variant of interest in computational complexity is "prefix induction", in which one needs to prove
forall k (P(k) to P(2k) land P(2k+1))
or equivalently
forall k left( P left(leftlfloor frac{k}{2} rightrfloor right) to P(k) right)
The induction principle then "automates" log n applications of this inference in getting from P(0) to P(n). (It is called "prefix induction" because each step proves something about a number from something about the "prefix" of that number formed by truncating the low bit of its binary representation. It can be viewed as an application of traditional induction on the length of that binary representation.)If traditional predecessor induction is interpreted computationally as an n-step loop, prefix induction corresponds to a log n-step loop, and thus proofs using prefix induction are "more feasibly constructive" than proofs using predecessor induction.Predecessor induction can trivially simulate prefix induction on the same statement. Prefix induction can simulate predecessor induction, but only at the cost of making the statement more syntactically complex (adding a bounded universal quantifier), so the interesting results relating prefix induction to polynomial-time computation depend on excluding unbounded quantifiers entirely, and limiting the alternation of bounded universal and existential quantifiers allowed in the statement.BOOK, Buss, Samuel, Bounded Arithmetic, 1986, Bibliopolis, Naples, One can take the idea a step further: one must prove
forall k left( P left( leftlfloor sqrt{k} rightrfloor right) to P(k) right)
whereupon the induction principle "automates" log log n applications of this inference in getting from P(0) to P(n). This form of induction has been used, analogously, to study log-time parallel computation.{{citation needed|date=January 2018}}Complete (strong) inductionAnother variant, called complete induction, course of values induction or strong induction (in contrast to which the basic form of induction is sometimes known as weak induction) makes the inductive step easier to prove by using a stronger hypothesis: one proves the statement {{nowrap|P(m + 1)}} under the assumption that P(n) holds for all natural n less than {{nowrap|m + 1}}; by contrast, the basic form only assumes P(m). The name "strong induction" does not mean that this method can prove more than "weak induction", but merely refers to the stronger hypothesis used in the inductive step; in fact the two methods are equivalent, as explained below. In this form of complete induction one still has to prove the base case, P(0), and it may even be necessary to prove extra base cases such as P(1) before the general argument applies, as in the example below of the Fibonacci number Fn.Although the form just described requires one to prove the base case, this is unnecessary if one can prove P(m) (assuming P(n) for all lower n) for all {{nowrap|m â‰¥ 0}}. This is a special case of transfinite induction as described below. In this form the base case is subsumed by the case {{Nowrap|1=m = 0}}, where P(0) is proved with no other P(n) assumed;this case may need to be handled separately, but sometimes the same argument applies for m = 0 and {{nowrap|m > 0}}, making the proof simpler and more elegant.In this method it is, however, vital to ensure that the proof of P(m) does not implicitly assume that {{nowrap|m > 0}}, e.g. by saying "choose an arbitrary {{nowrap|n |