Mathematical induction

History

In 370 BC, Plato's Parmenides may have contained an early example of an implicit inductive proof. The earliest implicit traces of mathematical induction may be found in Euclid's proof that the number of primes is infinite and in Bhaskara's "cyclic method". 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.

None of these ancient mathematicians, however, explicitly stated the inductive 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 Traité du triangle arithmétique (1665). Another Frenchman, Fermat, made ample use of a related principle, indirect proof by infinite descent. The inductive 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, Augustus de Morgan, Charles Sanders Peirce, Giuseppe Peano, and Richard Dedekind.

Description

The 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:

1. The basis (base case): prove that the statement holds for the first natural number n. Usually, n = 0 or n = 1, rarely, n = –1 (although not a natural number, the extension of the natural numbers to –1 is still a well-ordered set).
2. The inductive step: prove that, if the statement holds for some natural number n, then the statement holds for n + 1.

The hypothesis in the inductive step that the statement holds for some n is called the induction hypothesis (or inductive hypothesis). To perform the inductive step, one assumes the induction hypothesis and then uses this assumption to prove the statement for n + 1.

Whether n = 0 or n = 1 depends on the definition of the natural numbers. If 0 is considered a natural number, as is common in the fields of combinatorics and mathematical logic, the base case is given by n = 0. If, on the other hand, 1 is taken as the first natural number, then the base case is given by n = 1.

Example

Mathematical induction can be used to prove that the following statement, P(n), holds for all natural numbers n.

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.

Basis: Show that the statement holds for n = 0.
P(0) amounts to the statement:

In the left-hand side of the equation, the only term is 0, and so the left-hand side is simply equal to 0.
In the right-hand side of the equation, 0·(0 + 1)/2 = 0.
The two sides are equal, so the statement is true for n = 0. Thus it has been shown that P(0) holds.

Inductive step: Show that if P(k) holds, then also 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 P(k + 1) holds, that is:

Using the induction hypothesis that P(k) holds, the left-hand side can be rewritten to:

Algebraically:

thereby showing that indeed P(k + 1) holds.

Since both the basis and the inductive step have been performed, by mathematical induction, the statement P(n) holds for all natural numbers n. Q.E.D.

Axiom of induction

Mathematical induction as an inference rule can be formalized as a second-order axiom. The axiom of induction is, in logical symbols,

where P is any predicate and k and n are both natural numbers.

In words, the basis P(0) and the inductive step (namely, that the inductive 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 basis 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.

Variants

In practice, proofs by induction are often structured differently, depending on the exact nature of the property to be proved.

Induction basis 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 b, then the proof by induction consists of:

1. Showing that the statement holds when n = b.
2. Showing that if the statement holds for n = m ≥ b then the same statement also holds for n = m + 1.

This can be used, for example, to show that n² ≥ 3n for n ≥ 3. A more substantial example is a proof that

In this way, one can prove that P(n) holds for all n ≥1, or even n ≥−5. This form of mathematical induction is actually a special case of the previous form because if the statement to be proved is P(n) then proving it with these two rules is equivalent with proving P(n + b) for all natural numbers n with the first two steps.

Induction basis equal to 2

In mathematics, many standard functions, including operations such as "+" and relations such as "=", are binary, meaning that they take two arguments. Often these functions possess properties that implicitly extend them to more than two arguments. For example, once addition a + b is defined and is known to satisfy the associativity property (a + b) + c = a + (b + c), then the ternary addition a + b + c makes sense, either as (a + b) + c or as a + (b + c). Similarly, many axioms and theorems in mathematics are stated only for the binary versions of mathematical operations and relations, and implicitly extend to higher-arity versions.

Suppose that one wishes to prove a statement about an n-ary operation implicitly defined from a binary operation, using mathematical induction on n. In this case it is natural to take 2 for the induction basis.

Example: product rule for the derivative

In this example, the binary operation in question is multiplication (of functions). The usual product rule for the derivative taught in calculus states:

or in logarithmic derivative form

This can be generalized to a product of n functions. One has

or in logarithmic derivative form

In each of the n terms of the usual form, just one of the factors is a derivative; the others are not.

When this general fact is proved by mathematical induction, the n = 0 case is trivial, ( 1 ) ′ = 0 (since the empty product is 1, and the empty sum is 0). The n = 1 case is also trivial, f 1 ′ = f 1 ′ . And for each n ≥ 3, the case is easy to prove from the preceding n − 1 case. The real difficulty lies in the n = 2 case, which is why that is the one stated in the standard product rule.

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 performs a basis step and an inductive step for n, and in each of those performs a basis step 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 descent

The method of infinite descent was one of Pierre de Fermat's favorites. This method of proof can assume several slightly different forms. For example, it might begin by showing that if a statement is true for a natural number n it must also be true for some smaller natural number m (m < n). Using mathematical induction (implicitly) with the inductive hypothesis being that the statement is false for all natural numbers less than or equal to m, one may conclude that the statement cannot be true for any natural number n.

Although this particular form of infinite-descent proof is clearly a mathematical induction, whether one holds all proofs "by infinite descent" to be mathematical inductions depends on how one defines the term "proof by infinite descent." One might, for example, use the term to apply to proofs in which the well-ordering of the natural numbers is assumed, but not the principle of induction. Such, for example, is the usual proof that 2 has no rational square root (see Infinite descent).

Prefix induction

The most common form of induction requires proving that

or equivalently

whereupon the induction principle "automates" n applications of this inference 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

or equivalently

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.)

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. See

One could take it a step farther to "prefix of prefix induction": one must prove

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 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 inductive step easier to prove by using a stronger hypothesis: one proves the statement P(m + 1) under the assumption that P(n) holds for all natural n less than 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 F_n.

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 m ≥ 0. This is a special case of transfinite induction as described below. In this form the base case is subsumed by the case 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 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 m > 0, e.g. by saying "choose an arbitrary n < 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 0 ≤ m ≤ 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 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).

Examples of complete induction

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

where F_n is the nth Fibonacci number, φ = (1 + √5)/2 (the golden ratio) and ψ = (1 − √5)/2 are the roots of the polynomial x² − x − 1. By using the fact that F_n+2 = F_n+1 + F_n for each n ∈ N, the identity above can be verified by direct calculation for 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 n = 0 and n = 1.

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", and assume that for a given m > 1 it 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 = 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; i.e. a product of primes.

Transfinite induction

The last two steps can be reformulated as one step:

1. Showing that if the statement holds for all n < m then the same statement also holds for n = m.

This form of mathematical 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.

This form of induction, when applied to ordinals (which form a well-ordered and hence well-founded class), is called transfinite induction. It is an important proof technique in set theory, topology and other fields.

Proofs by transfinite induction typically distinguish three cases:

1. when m is a minimal element, i.e. there is no element smaller than m
2. when m has a direct predecessor, i.e. the set of elements which are smaller than m has a largest element
3. when m has no direct predecessor, i.e. m is a so-called limit-ordinal

Strictly speaking, it is not necessary in transfinite induction to prove the basis, because it is a vacuous special case of the proposition that if P is true of all n < m, then P is true of m. It is vacuously true precisely because there are no values of n < m that could serve as counterexamples.

Equivalence with the well-ordering principle

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:

To derive simple induction from these axioms, one must show that if P(n) is some proposition predicated of n, and if:

then P(n) holds for all n.

Proof. Let S be the set of all natural numbers for which P(n) is false. Let us see what happens if one asserts that S is nonempty. Well-ordering tells us that S has a least element, say t. Moreover, since P(0) is true, t is not 0. Since every natural number is either zero or some n + 1, there is some natural number n such that n + 1 = t. Now n is less than t, and t is the least element of S. It follows that n is not in S, and so P(n) is true. This means that P(n + 1) is true, and so P(t) is true. This is a contradiction, since t was in S. Therefore, S is empty.

It can also be proved that induction, given the other axioms, implies the well-ordering principle.

Proof. Suppose there exists a non-empty set, S, of naturals with no least element. Let P(k) be the assertion that k is not in S. Then P(0) is true for if it were false then 0 is the least element of S. Furthermore, suppose P(1), P(2),..., P(k) is true. Then if P(k+1) is false k+1 is in S, thus it is the minimal element is S, a contradiction. Thus P(k+1) is true. Therefore, by the induction axiom S is empty, a contradiction.

Example of error in the inductive step

This example demonstrated a subtle error in the proof of the inductive step.

Joel E. Cohen proposed the following argument, which purports to prove by mathematical induction that all horses are of the same color:

The basis case n = 1 is trivial (as any horse is the same color as itself), and the inductive step is correct in all cases n > 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).