CSNB143 – Discrete Structure Topic 5 – Induction Part I

Topic 5 – Induction Learning Outcomes Students should be able to explain the meaning of Principle of Mathematical Induction Students should be able to demonstrate each step involved in different type of induction

Topic 5 – Induction Introduction Mathematical Induction can be used to prove statements which asserts the propositional function P(x) is true for all positive integers x. Why do we need proof by induction? – Often theorems state that a propositional function, P(n) is true for all positive integers n – We prove the theorem by using mathematical induction Induction is the process by which we conclude that what is true of certain individuals of a class, is true of the whole class, or that what is true at certain times will be true in similar circumstances at all times

Topic 5 – Induction Introduction When we prove statements using mathematical induction, we first show that P(1) is true. Then we will check to make sure that P(2) is also true and P(3) is also true because P(2) implies (P3). Ways to remember:

Topic 5 – Induction Proof by Mathematical Induction A proof by mathematical induction has three parts 1.Basis Step 2.Inductive Step 1. Basic Step: Show that P(1) is true 2. Inductive Step: Assume P(k) is true and Show that P(k + 1) is true on the basis of the inductive hypothesis

Topic 5 – Induction Example (Summation Type) Show by mathematical induction, for all n  1; … + n = n (n+1) 2 1.Basic step Prove that P(1) is true. The first number in the sequence is 1, so P(1) = 1 (1 + 1)= 1 2 Therefore, it is true. ( can proceed to the next step)

Proving for all n  1; … + n = n (n+1) 2 Topic 5 – Induction We will at this stage ASSUME this is TRUE

Topic 5 – Induction Now check against the sequence to prove that after adding k+1 to k, the statement is still true

Topic 5 – Induction Conclusion So, with Principle of Mathematical Induction, P(n) is true for all n  1.