mathematical induction

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Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ...  all hold. Informal metaphors help to explain this technique, such as falling dominoes or climbing a ladder: A proof by induction consists of two cases. The first, the base case, proves the statement for ''n'' = 0 without assuming any knowledge of other cases. The second case, the induction step, proves that ''if'' the statement holds for any given case ''n'' = ''k'', ''then'' it must also hold for the next case ''n'' = ''k'' + 1. These two steps establish that the statement holds for every natural number ''n''. The base case does not necessarily begin with ''n'' = 0, but often with ''n'' = 1, and possibly with any fixed natural number ''n'' = ''N'', establishing the truth of the statement for all natural numbers ''n'' ≥ ''N''. The method can be extended to prove statements about more general
well-founded In mathematics, a binary relation ''R'' is called well-founded (or wellfounded) on a class ''X'' if every non-empty subset ''S'' ⊆ ''X'' has a minimal element with respect to ''R'', that is, an element ''m'' not related by ''s& ...
structures, such as
trees In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are u ...
; this generalization, known as
structural induction Structural induction is a proof method that is used in mathematical logic (e.g., in the proof of Łoś' theorem), computer science, graph theory, and some other mathematical fields. It is a generalization of mathematical induction over natural nu ...
, is used in
mathematical logic Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of for ...
and
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
. Mathematical induction in this extended sense is closely related to
recursion Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathemati ...
. Mathematical induction is an
inference rule In the philosophy of logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion (or conclusions). For example, the rule of ...
used in
formal proof In logic and mathematics, a formal proof or derivation is a finite sequence of sentences (called well-formed formulas in the case of a formal language), each of which is an axiom, an assumption, or follows from the preceding sentences in the seq ...
s, and is the foundation of most correctness proofs for computer programs. Although its name may suggest otherwise, mathematical induction should not be confused with inductive reasoning as used in
philosophy Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. Some ...
(see Problem of induction). The mathematical method examines infinitely many cases to prove a general statement, but does so by a finite chain of
deductive reasoning Deductive reasoning is the mental process of drawing deductive inferences. An inference is deductively valid if its conclusion follows logically from its premises, i.e. if it is impossible for the premises to be true and the conclusion to be fals ...
involving the
variable Variable may refer to: * Variable (computer science), a symbolic name associated with a value and whose associated value may be changed * Variable (mathematics), a symbol that represents a quantity in a mathematical expression, as used in many ...
''n'', which can take infinitely many values.

History

In 370 BC,
Plato Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
's Parmenides may have contained traces of an early example of an implicit inductive proof. An opposite iterated technique, counting ''down'' rather than up, is found in the
sorites paradox The sorites paradox (; sometimes known as the paradox of the heap) is a paradox that results from vague predicates. A typical formulation involves a heap of sand, from which grains are removed individually. With the assumption that removing a sing ...
, 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. The earliest implicit proof by mathematical induction is in the ''al-Fakhri'' written by al-Karaji around 1000 AD, who applied it to arithmetic sequences to prove the
binomial theorem In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial into a sum involving terms of the form , where the ...
and properties of
Pascal's triangle In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although o ...
. As Katz says In India, early implicit proofs by mathematical induction appear in Bhaskara's " cyclic method". 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 Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric. Odd may also refer to: Acronym * ODD (Text Encoding Initiative) ("One Document Does it all"), an abstracted literate-programming format for describing X ...
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s is ''n''2. The earliest
rigorous Rigour (British English) or rigor (American English; American and British English spelling differences#-our, -or, see spelling differences) describes a condition of stiffness or strictness. These constraints may be environmentally imposed, su ...
use of induction was by
Gersonides Levi ben Gershon (1288 – 20 April 1344), better known by his Graecized name as Gersonides, or by his Latinized name Magister Leo Hebraeus, or in Hebrew by the abbreviation of first letters as ''RaLBaG'', was a medieval French Jewish philosoph ...
(1288–1344). The first explicit formulation of the principle of induction was given by Pascal in his ''Traité du triangle arithmétique'' (1665). Another Frenchman,
Fermat Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is ...
, made ample use of a related principle: indirect proof by
infinite descent In mathematics, a proof by infinite descent, also known as Fermat's method of descent, is a particular kind of proof by contradiction used to show that a statement cannot possibly hold for any number, by showing that if the statement were to hold f ...
. The induction hypothesis was also employed by the Swiss
Jakob Bernoulli Jacob Bernoulli (also known as James or Jacques; – 16 August 1705) was one of the many prominent mathematicians in the Bernoulli family. He was an early proponent of Leibnizian calculus and sided with Gottfried Wilhelm Leibniz during the Leib ...
, and from then on it became well known. The modern formal treatment of the principle came only in the 19th century, with
George Boole George Boole (; 2 November 1815 – 8 December 1864) was a largely self-taught English mathematician, philosopher, and logician, most of whose short career was spent as the first professor of mathematics at Queen's College, Cork in ...
, Augustus de Morgan,
Charles Sanders Peirce Charles Sanders Peirce ( ; September 10, 1839 – April 19, 1914) was an American philosopher, logician, mathematician and scientist who is sometimes known as "the father of pragmatism". Educated as a chemist and employed as a scientist for t ...
,
Giuseppe Peano Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The sta ...
, and Richard Dedekind.

Description

The simplest and most common form of mathematical induction infers that a statement involving a
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
(that is, an integer or 1) holds for all values of . The proof consists of two steps: # The base case (or initial case): prove that the statement holds for 0, or 1. # The induction step (or inductive step, or step case): prove that for every , if the statement holds for , then it holds for . In other words, assume that the statement holds for some arbitrary natural number , and prove that the statement holds for . The hypothesis in the induction step, that the statement holds for a particular , is called the induction hypothesis or inductive hypothesis. To prove the induction step, one assumes the induction hypothesis for and then uses this assumption to prove that the statement holds for . Authors who prefer to define natural numbers to begin at 0 use that value in the base case; those who define natural numbers to begin at 1 use that value.

Examples

Sum of consecutive natural numbers

Mathematical induction can be used to prove the following statement ''P''(''n'') for all natural numbers ''n''. : $P\left(n\right)\!:\ \ 0 + 1 + 2 + \cdots + n = \frac.$ This states a general formula for the sum of the natural numbers less than or equal to a given number; in fact an infinite sequence of statements: $0 = \tfrac2$, $0+1 = \tfrac2$, $0+1+2 = \tfrac2$, etc. Proposition. For every $n\in\mathbb$, $0 + 1 + 2 + \cdots + n = \tfrac.$ Proof. Let ''P''(''n'') be the statement $0 + 1 + 2 + \cdots + n = \tfrac.$ We give a proof by induction on ''n''. ''Base case:'' Show that the statement holds for the smallest natural number ''n'' = 0. ''P''(0) is clearly true: $0 = \tfrac\,.$ ''Induction step:'' Show that for every ''k ≥'' 0, if ''P''(''k'') holds, then ''P''(''k'' + 1) also holds. Assume the induction hypothesis that for a particular ''k'', the single case ''n'' = ''k'' holds, meaning ''P''(''k'') is true:
$0 + 1 + \cdots + k = \frac2.$
It follows that: : $\left(0 + 1 + 2 + \cdots + k \right)+ \left(k+1\right) = \frac2 + \left(k+1\right).$
Algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
ically, the right hand side simplifies as: : $\begin \frac + \left(k+1\right) &= \frac \\ &= \frac \\ &= \frac. \end$ Equating the extreme left hand and right hand sides, we deduce that:
$0 + 1 + 2 + \cdots + k + \left(k+1\right) = \frac2.$
That is, the statement ''P''(''k'' + 1) also holds true, establishing the induction step. ''Conclusion:'' Since both the base case and the induction step have been proved as true, by mathematical induction the statement ''P''(''n'') holds for every natural number ''n''.

A trigonometric inequality

Induction is often used to prove
inequalities Inequality may refer to: Economics * Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy * Economic inequality, difference in economic well-being between population groups * ...
. As an example, we prove that $, \!\sin nx, \leq n\,, \!\sin x,$ for any
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
$x$ and natural number $n$. At first glance, it may appear that a more general version, $, \!\sin nx, \leq n\,, \!\sin x,$ for any ''real'' numbers $n,x$, could be proven without induction; but the case $n=\frac,\, x=\pi$ shows it may be false for non-integer values of $n$. This suggests we examine the statement specifically for ''natural'' values of $n$, and induction is the readiest tool. Proposition. For any $x \in \mathbb$ and $n \in \mathbb$, $, \!\sin nx, \leq n\,, \!\sin x,$. Proof. Fix an arbitrary real number $x$, and let $P\left(n\right)$ be the statement $, \!\sin nx, \leq n\,, \!\sin x,$. We induct on $n$. ''Base case:'' The calculation $, \!\sin 0x, = 0 \leq 0 = 0\,, \!\sin x,$ verifies $P\left(0\right)$. ''Induction step:'' We show the implication $P\left(k\right) \implies P\left(k+1\right)$ for any natural number $k$. Assume the induction hypothesis: for a given value $n = k \geq 0$, the single case $P\left(k\right)$ is true. Using the angle addition formula and the
triangle inequality In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but ...
, we deduce: : $\begin , \!\sin\left(k+1\right)x, &= & , \!\sin kx\,\cos x+\sin x\,\cos kx\,, & \text \\ &\leq & , \!\sin kx\,\cos x, + , \!\sin x\,\cos kx, & \text \\ &= & , \!\sin kx, \,, \!\cos x, +, \!\sin x, \,, \!\cos kx, & \\ &\leq & , \!\sin kx, + , \!\sin x, & \left(, \!\cos t, \leq 1\right) \\ &\leq & k\,, \!\sin x, +, \!\sin x, & \text\right)\\ &= & \left(k+1\right)\,, \!\sin x, . & \end$ The inequality between the extreme left-hand and right-hand quantities shows that $P\left(k+1\right)$ is true, which completes the induction step. ''Conclusion:'' The proposition $P\left(n\right)$ holds for all natural numbers $n$. ∎

Variants

In practice, proofs by induction are often structured differently, depending on the exact nature of the property to be proven. All variants of induction are special cases of
transfinite induction Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Its correctness is a theorem of ZFC. Induction by cases Let P(\alpha) be a property defined for ...
; see below.

Base case other than 0 or 1

If one wishes to prove a statement, not for all natural numbers, but only for all numbers greater than or equal to a certain number , then the proof by induction consists of the following: # Showing that the statement holds when . # Showing that if the statement holds for an arbitrary number , then the same statement also holds for . This can be used, for example, to show that for . In this way, one can prove that some statement holds for all , or even for all . This form of mathematical induction is actually a special case of the previous form, because if the statement to be proved is then proving it with these two rules is equivalent with proving for all natural numbers with an induction base case .

Example: forming dollar amounts by coins

Assume an infinite supply of 4- and 5-dollar coins. Induction can be used to prove that any whole amount of dollars greater than or equal to can be formed by a combination of such coins. Let denote the statement " dollars can be formed by a combination of 4- and 5-dollar coins". The proof that is true for all can then be achieved by induction on as follows: ''Base case:'' Showing that holds for is simple: take three 4-dollar coins. ''Induction step:'' Given that holds for some value of (''induction hypothesis''), prove that holds, too. Assume is true for some arbitrary . If there is a solution for dollars that includes at least one 4-dollar coin, replace it by a 5-dollar coin to make dollars. Otherwise, if only 5-dollar coins are used, must be a multiple of 5 and so at least 15; but then we can replace three 5-dollar coins by four 4-dollar coins to make dollars. In each case, is true. Therefore, by the principle of induction, holds for all , and the proof is complete. In this example, although also holds for $k \in \$, the above proof cannot be modified to replace the minimum amount of dollar to any lower value . For , the base case is actually false; for , the second case in the induction step (replacing three 5- by four 4-dollar coins) will not work; let alone for even lower .

Induction on more than one counter

It 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 induction step for ''n'', and in each of those proves a base case and an induction 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 descent

The method of infinite descent is a variation of mathematical induction which was used by
Pierre de Fermat Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he ...
. 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, thereby 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 induction

The most common form of proof by mathematical induction requires proving in the induction step that : $\forall k \left(P\left(k\right) \to P\left(k+1\right)\right)$ 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 proves the following statement in the induction step: : $\forall k \left(P\left(k\right) \to P\left(2k\right) \land P\left(2k+1\right)\right)$ or equivalently : $\forall k \left\left( P\!\left\left(\left\lfloor \frac \right\rfloor \right\right) \to P\left(k\right) \right\right)$ The induction principle then "automates" log2 ''n'' applications of this inference in getting from ''P''(0) to ''P''(''n''). In fact, it is called "prefix induction" because each step proves something about a number from something about the "prefix" of that number — as formed by truncating the low bit of its
binary representation A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method of mathematical expression which uses only two symbols: typically "0" (zero) and "1" (one). The base-2 numeral system is a positional notation ...
. It can also 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, then prefix induction would correspond to a log-''n''-step loop. Because of that, 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 Existentialism ( ) is a form of philosophical inquiry that explores the problem of human existence and centers on human thinking, feeling, and acting. Existentialist thinkers frequently explore issues related to the meaning, purpose, and valu ...
quantifiers allowed in the statement. One can take the idea a step further: one must prove : $\forall k \left\left( P\!\left\left( \left\lfloor \sqrt \right\rfloor \right\right) \to P\left(k\right) \right\right)$ 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.

Complete (strong) induction

Another 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 induction step easier to prove by using a stronger hypothesis: one proves the statement $P\left(m+1\right)$ under the assumption that $P\left(n\right)$ holds for ''all'' natural numbers $n$ less than $m+1$; by contrast, the basic form only assumes $P\left(m\right)$. 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 induction step. In fact, it can be shown that the two methods are actually equivalent, as explained below. In this form of complete induction, one still has to prove the base case, $P\left(0\right)$, and it may even be necessary to prove extra-base cases such as $P\left(1\right)$ before the general argument applies, as in the example below of the
Fibonacci number In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...
$F_$. Although the form just described requires one to prove the base case, this is unnecessary if one can prove $P\left(m\right)$ (assuming $P\left(n\right)$ for all lower $n$) for all $m\geq 0$. This is a special case of
transfinite induction Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Its correctness is a theorem of ZFC. Induction by cases Let P(\alpha) be a property defined for ...
as described below, although it is no longer equivalent to ordinary induction. In this form the base case is subsumed by the case $m=0$, where $P\left(0\right)$ is proved with no other $P\left(n\right)$ assumed; this case may need to be handled separately, but sometimes the same argument applies for $m=0$ and $m>0$, making the proof simpler and more elegant. In this method, however, it is vital to ensure that the proof of $P\left(m\right)$ does not implicitly assume that $m>0$, e.g. by saying "choose an arbitrary
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is b ...
$P\left(n\right)$ holds for all $n$, and our proof of $P\left(n\right)$ is easily transformed into a proof of $Q\left(n\right)$ by (ordinary) induction. If, on the other hand, $P\left(n\right)$ had been proven by ordinary induction, the proof would already effectively be one by complete induction: $P\left(0\right)$ is proved in the base case, using no assumptions, and $P\left(n+1\right)$ is proved in the induction step, in which one may assume all earlier cases but need only use the case $P\left(n\right)$.

Example: Fibonacci numbers

Complete induction is most useful when several instances of the inductive hypothesis are required for each induction step. For example, complete induction can be used to show that : $F_n = \frac$ where $F_n$ is the ''n''th
Fibonacci number In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...
, and $\varphi =$ (the
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...
) and $\psi =$ are the
roots A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients. Root or roots may also refer to: Art, entertainment, and media * ''The Root'' (magazine), an online magazine focusing ...
of the
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
$x^2-x-1$. By using the fact that $F_ = F_ + F_$ for each $n \in \Bbb$, the identity above can be verified by direct calculation for $F_$ if one assumes that it already holds for both $F_$ and $F_n$. To complete the proof, the identity must be verified in the two base cases: $n=0$ and $n=1$.

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 In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
greater than 1 is a product of (one or more)
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s", which is the "
existence Existence is the ability of an entity to interact with reality. In philosophy, it refers to the ontological property of being. Etymology The term ''existence'' comes from Old French ''existence'', from Medieval Latin ''existentia/exsistentia' ...
" part of the
fundamental theorem of arithmetic In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the ord ...
. For proving the induction step, the induction hypothesis is that for a given $n>1$ the statement holds for all smaller $n>1$. If $m$ is prime then it is certainly a product of primes, and if not, then by definition it is a product: $m=n_1 n_2$, where neither of the factors is equal to 1; hence neither is equal to $m$, and so both are greater than 1 and smaller than $m$. The induction hypothesis now applies to $n_1$ and $n_2$, so each one is a product of primes. Thus $m$ is a product of products of primes, and hence by extension a product of primes itself.

Example: dollar amounts revisited

We shall look to prove the same example as above, this time with ''strong induction''. The statement remains the same: : $S\left(n\right): \,\,n \geq 12 \implies \,\exists\, a,b\in\mathbb. \,\, n = 4a+5b$ However, there will be slight differences in the structure and the assumptions of the proof, starting with the extended base case. Proof. ''Base case:'' Show that $S\left(k\right)$ holds for $k = 12,13,14,15$. : $\begin 4 \cdot 3+5 \cdot 0=12\\ 4 \cdot 2+5 \cdot 1=13\\ 4 \cdot 1+5 \cdot 2=14\\ 4 \cdot 0+5 \cdot 3=15 \end$ The base case holds. ''Induction step:'' Given some $j>15$, assume $S\left(m\right)$ holds for all $m$ with $12\leq m< j$. Prove that $S\left(j\right)$ holds. Choosing $m=j-4$, and observing that

Forward-backward induction

Sometimes, it is more convenient to deduce backwards, proving the statement for $n-1$, given its validity for $n$. However, proving the validity of the statement for no single number suffices to establish the base case; instead, one needs to prove the statement for an infinite subset of the natural numbers. For example,
Augustin Louis Cauchy Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...
first used forward (regular) induction to prove the
inequality of arithmetic and geometric means In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM–GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; and ...
for all
powers of 2 A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer  as the exponent. In a context where only integers are considered, is restricted to non-negative ...
, and then used backwards induction to show it for all natural numbers.

Example of error in the induction step

The induction step must be proved for all values of ''n''. To illustrate this, Joel E. Cohen proposed the following argument, which purports to prove by mathematical induction that all horses are of the same color:. Reprinted in ''A Random Walk in Science'' (R. L. Weber, ed.), Crane, Russak & Co., 1973. ''Base case:'' in a set of only ''one'' horse, there is only one color. ''Induction step:'' assume as induction hypothesis that within any set of $n$ horses, there is only one color. Now look at any set of $n+1$ horses. Number them: $1, 2, 3, \dotsc, n, n+1$. Consider the sets $\left\$ and $\left\$. Each is a set of only $n$ horses, therefore within each there is only one color. But the two sets overlap, so there must be only one color among all $n+1$ horses. The base case $n=1$ is trivial, and the induction step is correct in all cases $n > 1$. However, the argument used in the induction step is incorrect for $n+1=2$, because the statement that "the two sets overlap" is false for $\left\$ and $\left\$.

Formalization

In
second-order logic In logic and mathematics, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory. First-order logic quantifies on ...
, one can write down the "
axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
of induction" as follows: : $\forall P\Bigl\left( P\left(0\right) \land \forall k \bigl\left( P\left(k\right) \to P\left(k+1\right)\bigr \right) \to \forall n \bigl\left(P\left(n\right)\bigr\right)\Bigr\right)$, where ''P''(.) is a variable for predicates involving one natural number and ''k'' and ''n'' are variables for
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
s. In words, the base case and the induction step (namely, that the induction hypothesis implies ) together imply that for any natural number . The axiom of induction asserts the validity of inferring that holds for any natural number from the base case and the induction step. 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 In logic and mathematics, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory. First-order logic quantifies on ...
. Axiomatizing arithmetic induction in
first-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
requires an
axiom schema In mathematical logic, an axiom schema (plural: axiom schemata or axiom schemas) generalizes the notion of axiom. Formal definition An axiom schema is a formula in the metalanguage of an axiomatic system, in which one or more schematic variables ap ...
containing a separate axiom for each possible predicate. The article
Peano axioms In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly ...
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 the following four other axioms: # 0 is a natural number. # The successor function of every natural number yields a natural number . # The successor function is injective. # 0 is not in the
range Range may refer to: Geography * Range (geographic), a chain of hills or mountains; a somewhat linear, complex mountainous or hilly area (cordillera, sierra) ** Mountain range, a group of mountains bordered by lowlands * Range, a term used to i ...
of . In
first-order In mathematics and other formal sciences, first-order or first order most often means either: * "linear" (a polynomial of degree at most one), as in first-order approximation and other calculus uses, where it is contrasted with "polynomials of high ...
ZFC set theory, quantification over predicates is not allowed, but one can still express induction by quantification over sets: : $\forall A \Bigl\left( 0 \in A \land \forall k \in \N \bigl\left( k \in A \to \left(k+1\right) \in A \bigr\right) \to \N\subseteq A\Bigr\right)$ 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

One variation of the principle of complete induction can be generalized for statements about elements of any well-founded set, that is, a set with an
irreflexive relation In mathematics, a binary relation ''R'' on a set ''X'' is reflexive if it relates every element of ''X'' to itself. An example of a reflexive relation is the relation " is equal to" on the set of real numbers, since every real number is equal to ...
< that contains no
infinite descending chain In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some Set (mathematics), set X, which satisfies the following for all a, b and c in X: # a ...
s. Every set representing an ordinal number is well-founded, the set of natural numbers is one of them. Applied to a well-founded set, transfinite induction can be formulated as a single step. To prove that a statement ''P''(''n'') holds for each ordinal number: # Show, for each ordinal number ''n'', that if ''P''(''m'') holds for all , then ''P''(''n'') also holds. This form of induction, when applied to a set of ordinal numbers (which form a
well-order In mathematics, a well-order (or well-ordering or well-order relation) on a set ''S'' is a total order on ''S'' with the property that every non-empty subset of ''S'' has a least element in this ordering. The set ''S'' together with the well-or ...
ed and hence well-founded
class Class or The Class may refer to: Common uses not otherwise categorized * Class (biology), a taxonomic rank * Class (knowledge representation), a collection of individuals or objects * Class (philosophy), an analytical concept used differentl ...
), is called ''
transfinite induction Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Its correctness is a theorem of ZFC. Induction by cases Let P(\alpha) be a property defined for ...
''. It is an important proof technique in
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
,
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
and other fields. Proofs by transfinite induction typically distinguish three cases: # 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 In set theory, a limit ordinal is an ordinal number that is neither zero nor a successor ordinal. Alternatively, an ordinal λ is a limit ordinal if there is an ordinal less than λ, and whenever β is an ordinal less than λ, then there exists a ...
. Strictly speaking, it is not necessary in transfinite induction to prove a base case, because it is a vacuous special case of the proposition that if ''P'' is true of all , then ''P'' is true of ''m''. It is vacuously true precisely because there are no values of that could serve as counterexamples. So the special cases are special cases of the general case.

Relationship to the well-ordering principle

The principle of mathematical induction is usually stated as an
axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
of the natural numbers; see
Peano axioms In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly ...
. It is strictly stronger than the
well-ordering principle In mathematics, the well-ordering principle states that every non-empty set of positive integers contains a least element. In other words, the set of positive integers is well-ordered by its "natural" or "magnitude" order in which x precedes y i ...
in the context of the other Peano axioms. Suppose the following: * The trichotomy axiom: For any natural numbers ''n'' and ''m'', ''n'' is less than or equal to ''m'' if and only if ''m'' is not less than ''n''. * For any natural number ''n'', is greater . * For any natural number ''n'', no natural number is and . * No natural number is less than zero. It can then be proved that induction, given the above-listed axioms, implies the well-ordering principle. The following proof uses complete induction and the first and fourth axioms. Proof. Suppose there exists a
non-empty In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other t ...
set, ''S'', of natural numbers that has no least element. Let ''P''(''n'') be the assertion that ''n'' is not in ''S''. Then ''P''(0) is true, for if it were false then 0 is the least element of ''S''. Furthermore, let ''n'' be a natural number, and suppose ''P''(''m'') is true for all natural numbers ''m'' less than . Then if is false is in ''S'', thus being a minimal element in ''S'', a contradiction. Thus is true. Therefore, by the complete induction principle, ''P''(''n'') holds for all natural numbers ''n''; so ''S'' is empty, a contradiction. ∎ On the other hand, the set $\ \cup \$, shown in the picture, is well-ordered by the
lexicographic order In mathematics, the lexicographic or lexicographical order (also known as lexical order, or dictionary order) is a generalization of the alphabetical order of the dictionaries to sequences of ordered symbols or, more generally, of elements of a ...
. Moreover, except for the induction axiom, it satisfies all Peano axioms, where Peano's constant 0 is interpreted as the pair (0, 0), and Peano's ''successor'' function is defined on pairs by for all $x \in \$ and $n \in \mathbb$. As an example for the violation of the induction axiom, define the predicate as or for some $y \in \$ and $m \in \mathbb$. Then the base case ''P''(0, 0) is trivially true, and so is the induction step: if , then . However, ''P'' is not true for all pairs in the set. Peano's axioms with the induction principle uniquely model the natural numbers. Replacing the induction principle with the well-ordering principle allows for more exotic models that fulfill all the axioms. It is mistakenly printed in several books and sources that the well-ordering principle is equivalent to the induction axiom. In the context of the other Peano axioms, this is not the case, but in the context of other axioms, they are equivalent; specifically, the well-ordering principle implies the induction axiom in the context of the first two above listed axioms and * Every natural number is either 0 or for some natural number . A common mistake in many erroneous proofs is to assume that is a unique and well-defined natural number, a property which is not implied by the other Peano axioms.

*
Combinatorial proof In mathematics, the term ''combinatorial proof'' is often used to mean either of two types of mathematical proof: * A proof by double counting. A combinatorial identity is proven by counting the number of elements of some carefully chosen set in t ...
*
Induction puzzles Induction puzzles are logic puzzles, which are examples of multi-agent reasoning, where the solution evolves along with the principle of induction. A puzzle's scenario always involves multiple players with the same reasoning capability, who go ...
*
Proof by exhaustion Proof by exhaustion, also known as proof by cases, proof by case analysis, complete induction or the brute force method, is a method of mathematical proof in which the statement to be proved is split into a finite number of cases or sets of equiv ...
*
Recursion Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathemati ...
* Recursion (computer science) *
Structural induction Structural induction is a proof method that is used in mathematical logic (e.g., in the proof of Łoś' theorem), computer science, graph theory, and some other mathematical fields. It is a generalization of mathematical induction over natural nu ...
*
Transfinite induction Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Its correctness is a theorem of ZFC. Induction by cases Let P(\alpha) be a property defined for ...

References

Introduction

* (Ch. 8.) * * * (Section 1.2.1: Mathematical Induction, pp. 11–21.) * (Section 3.8: Transfinite induction, pp. 28–29.)

History

* * * * * * * * Reprinted (CP 3.252–288), (W 4:299–309) * * * * * * * * * {{Authority control Articles containing proofs