Mathematical Induction: Proof for Universal Statements on Natural Numbers (90 characters), Schemes and Mind Maps of Discrete Mathematics

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Natural Numbers:

Mathematical Induction

 Mathematical proof method that is used to prove a given statement about any well- organized set. Important proof technique for proving such universally quantified statements

EXAMPLE

 Example 1: For all n ≥ 1, prove that, 12 + 2^2 + 3^2 ….n^2 = {n(n + 1) (2n + 1)} / 6  Solution: Let the given statement be P(n), Now, let’s take a positive integer, k, and assume P(k) to be true i.e., We shall now prove that P(k + 1) is also true, so now we have, P(k + 1) = P(k) + (k + 1)^2

Direct Proof (Proof by Construction)

 In a constructive proof one attempts to demonstrate P ⇒ Q directly.  (^) There are only two steps to a direct proof (the second step is, of course, the tricky part):

1. Assume that P is true.
2. Use P to show that Q must be true.