Instructor: Is l Dillig, CS311H: Discrete Mathematics Mathematical Induction 10/26 Example 4 I Prove that 3 j (n 3 n ) for all positive integers n . You start off with a proof that the result holds for 0. Step 1 is usually easy, we just have to prove it is true for n=1. In the world of numbers we say: Step 1. That is how Mathematical Induction works. Step 2 is best done this way: Assume it is true for n=k 2 CS 441 Discrete mathematics for CS M. Hauskrecht Mathematical induction • Used to prove statements of the form x P(x) where x Z+ Mathematical induction proofs consists of two steps: 1) Basis: The proposition P(1) is true. Mathematical Induction - Problems With Solutions Several problems with detailed solutions on mathematical induction are presented. 2) Inductive Step: The implication P(n) P(n+1), is true for all positive n. The principle of mathematical induction is used to prove that a given proposition (formula, equality, inequality…) is true for all positive integer numbers greater than or equal to some integer N. Show it is true for first case, usually n=1; Step 2. An Analogy: A proof by mathematical induction is similar to knocking over a row of closely spaced dominos that are standing on end.To knock over the dominos in Figure 3.7.2, all you need to do is push the first domino over.To be assured that they all will be knocked over, some work must be done ahead of time. Since this is a discrete math for computer science course, I often continue onward by talking about induction as a "machine." Show that if n=k is true then n=k+1 is also true; How to Do it.