Changes: Mathematical induction

Revision as of 22:56, 9 August 2009

Mathematical induction is a method of proof by which a statement about a variable can be demonstrated to be true for all integer values of that variable greater than or equal to a specified integer (usually 0 or 1).

An example of such a statement is:

The number of possible pairings of n distinct objects is $\frac{n(n+1)}{2}$ (for any positive integer n).

A proof by induction proceeds as follows:

The statement is proved for the first possible value of n (usually 0 or 1, but other "starting values" are possible).
The statement is assumed to be true for some fixed, but unspecified, value n and this assumption is used to prove that the statement is true for $n+1$ (the latter statement is simply the original statement with n replaced by $n+1$ ).
The statement is then concluded to be true for all relevant values of n (all nonnegative values or all positive values, depending).

That the conclusion in step 3 above follows from steps 1 and 2 is the principle of mathematical induction.

More formally, given a proposition $P(n)$ about the variable n that is to be proved for $n \geq n_0$ , the following must be proved.

$P(n_0)$
$P(n) \implies P(n+1)$

The conclusion is then

$P(n)$ for $n \geq n_0$ .

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 '''Mathematical induction''' is a method of [[proof]] by which a statement about a [[variable]] can be demonstrated to be true for all [[integer]] values of that variable greater than or equal to a specified integer (usually 0 or 1).
-Examples of such statements include:
+An example of such a statement is:
 * The number of possible pairings of ''n'' distinct objects is <math>\frac{n(n+1)}{2}</math> (for any [[positive]] integer ''n'').
-* The difference between the [[square (number)|square]]s of any two [[consecutive]] [[whole number]]s is an [[odd number]].
-*: (This can be rephrased as: <math>(n+1)^2-n^2</math> is odd for any nonnegative integer ''n''.)
 A '''proof by induction''' proceeds as follows: