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Induction and Summation

Let P(n) be the property that

displaymath270

We use induction to prove that P(n) is true for all integers tex2html_wrap_inline304 .

Basis step. If n=1, then

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Inductive step. Assume as induction hypothesis that P(n) is true for some arbitrary, but fixed integer tex2html_wrap_inline310 . i.e.,

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We have to show that P(n+1) is also true, or equivalently

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We have

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Induction and Inequality

Let P(n) be the property

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We prove that P(n) is true for all tex2html_wrap_inline318 .

Basis step. If n=3, then

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Inductive step. Assume as induction hypothesis that P(n),

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is true for some arbitrary, but fixed integer tex2html_wrap_inline324 . We have to show that P(n+1),

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is also true.

We have

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Another Induction Proof

Let P(n) be the property

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where a and b are some nonnegative real numbers. We prove that P(n) is true for all tex2html_wrap_inline336 .

Basis step. If n=0 we have to prove

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which is true because tex2html_wrap_inline340 .

Inductive step. Assume as induction hypothesis that P(n),

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is true for some arbitrary, but fixed integer tex2html_wrap_inline344 . We have to show that P(n+1),

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is also true. To prove the latter implication, let us assume (1) tex2html_wrap_inline348 . We then need to show tex2html_wrap_inline350 .

From the induction hypothesis and (1) we may infer (2) tex2html_wrap_inline352 by modus ponens.

Since a is nonnegative, we get (3) tex2html_wrap_inline356 from (2).

Since tex2html_wrap_inline358 is nonnegative, we get (4) tex2html_wrap_inline360 from (1).

Putting (3) and (4) together, we obtain

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Strong Mathematical Induction

The inductive step of mathematical induction requires one to show that P(k+1) is true whenever P(k) is true. This can be generalized as follows.

Principle of Strong Mathematical Induction

Let P(n) be a predicate defined on the natural numbers and let a and b be natural numbers with tex2html_wrap_inline372 . Suppose the following two statements are true:
  1. tex2html_wrap_inline374 are true.
  2. For all natural numbers tex2html_wrap_inline376 , if P(i) is true for all natural numbers i with tex2html_wrap_inline382 , then P(k+1) is true.
Then P(n) is true for all natural numbers n with tex2html_wrap_inline390 .
Again the first part is called the basis step, and the second part, the inductive step.

This principle looks stronger than plain mathematical induction, because the inductive hypothesis

P(i) is true for all natural numbers i with tex2html_wrap_inline396 ,
is more general.

But one can show that strong mathematical induction can be derived from the standard mathematical induction principle.

Divisibility by a Prime

Let P(n) be the property

n is divisible by a prime number.
We prove that P(n) is true for all integers n with n> 1.

Basis step. If n=2, then P(n) is true because 2 is a prime and every number divides itself.

Inductive step. Let k be some integer with tex2html_wrap_inline416 and assume, as induction hypothesis, that P(i) is true for all integers i with tex2html_wrap_inline422 .

We have to show that P(k+1) is true, i.e., that k+1 is divisible by a prime.

We distinguish two cases.

Case (i). If k+1 is a prime, it is divisible by a prime (namely by itself).

Case (ii). If k+1 is not a prime, then it is a product tex2html_wrap_inline432 of two integers s and t, such that 1 < s < k+1 and 1 < t < k+1.

By the induction hypothesis, t is divisible by a prime. By the transitivity of divisibility, tex2html_wrap_inline444 is also divisible by a prime.

We have shown that in either case k+1 is divisible by a prime, which completes the proof. height6pt width4pt

The strong induction principle was applied here for values a=b=2.

Inductively Defined Structures

Many data structures are formally defined by mathematical induction. We have seen several examples, for instance, lists of elements of a certain type t.

Basis.

The empty list, denoted by [] or nil, is a t-list.

Induction.

If L is a t-list and a is an object of type t, then a::L is a t-list.

The key observation is that such a definition uniquely specifies a set of data objects and that the inherent inductive structure can be used to prove properties about the data type or functions defined on it.

Another example is the set of all strings that can be composed from characters of a given set tex2html_wrap_inline470 . This set tex2html_wrap_inline472 can be defined inductively as follows:

Basis.

The empty string, denoted by tex2html_wrap_inline474 , is an element of tex2html_wrap_inline476 .

Induction.

If a is a symbol in tex2html_wrap_inline480 and s is a string in tex2html_wrap_inline484 , then tex2html_wrap_inline486 is also a string in tex2html_wrap_inline488 .




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Next: About this document Up: My Home Page

Steve Skiena
Tue Aug 24 21:39:28 EDT 1999