csnb143 – discrete structure topic 5 – induction part i

CSNB143 – Discrete Structure

Topic 5 – Induction Part I

Topic 5 – InductionLearning 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 – InductionIntroduction• 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 – InductionIntroduction• 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 – InductionProof by Mathematical Induction• A proof by mathematical induction has three parts

1. Basis Step2. 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 – InductionExample (Summation Type)

• Show by mathematical induction, for all n 1; 1 + 2 + 3 + … + n = n (n+1) 21. 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; 1 + 2 + 3 + … + n = n (n+1) 2

Topic 5 – Induction

We will at this stage ASSUME this is TRUE

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

Topic 5 – Induction

ConclusionSo, with Principle of Mathematical Induction, P(n) is true for all n 1.