Finite Automata - PROOF TECHNIQUES, Study notes of Theory of Automata

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CHAPTER 7: PROOFCHAPTER 7: PROOF

TECHNIQUES TECHNIQUES

Fundamental Discrete Structure

BCT 1073

7.1 Introduction

7.2 Direct method

7.3 Indirect method

7.4 Contradiction method

7.5 Mathematical Induction

CONTENT

7.1: INTRODUCTION7.1: INTRODUCTION

Introduction

• A theorem is a statement that can be shown to be

true.

• Theorems can also be referred to as facts or results.
• A proof is a valid argument that established the truth of a theorem.
• A proof can include axioms (or postulate ), which are statements we assume to be true.

Applications

It is used to prove results about the:

• (^) complexity of algorithms
• (^) theorems about graphs and trees
• wide range of identities and inequalities.

Some Applications in

CS

• Verifying the correctness of certain types of

computer programs

• Establishing that operating system are secure
• Making inferences in Artificial Intelligence (AI)
• Showing that system specifications are consistent

Note that an integer is either even or odd , and no integer is both even and odd.

Note that an integer is either even or odd , and no integer is both even and odd.

The integer n is even if there exists an integer k such that n = 2 k.

The integer n is even if there exists an integer k such that n = 2 k.

The integer n is odd if there exists an integer k such that n = 2 k + 1.

The integer n is odd if there exists an integer k such that n = 2 k + 1.

Useful Definitions

7.2 : DIRECT METHOD7.2 : DIRECT METHOD

Lesson outcome:

Apply direct method to prove a theorem.

Give a direct proof of the theorem “If n is an odd integer, then n^2 is odd.”

We want to show that n^2 is an odd integer.

Let n is an odd integer:

Therefore, n^2 is an odd integer.

Example 7.2.

SOLUTIONS

Give a direct proof that if m and n are both perfect squares, then nm is also a perfect square. (An integer a is a perfect square if there is an integer b such that a = b^2 ).

We want to show that mn is perfect square.

Therefore, mn is perfect square.

Let m and n are both perfect squares:

Example 7.2.

SOLUTIONS

7.3: INDIRECT METHOD7.3: INDIRECT METHOD

Lesson outcome:

Apply indirect method to prove a

theorem.

 (^) Is known as proof by contraposition.

 (^) Let p  q ,

» (^) converse: q  p » (^) contrapositive: ~ q  ~ p » (^) inverse: ~ p  ~ q

 (^) Is known as proof by contraposition.

 (^) Let p  q ,

» (^) converse: q  p » (^) contrapositive: ~ q  ~ p » (^) inverse: ~ p  ~ q

INDIRECT METHOD

1. Use an indirect method of proof to show that if xy is odd integer, then x and y are also odd integers.
2. Show that if n is an integer and n^3 +5 is odd, then n is even using a proof by contraposition.
3. Use an indirect method of proof to show that if xy is odd integer, then x and y are also odd integers.
4. Show that if n is an integer and n^3 +5 is odd, then n is even using a proof by contraposition.

EXERCISE 7.

7.4: CONTRADICTION7.4: CONTRADICTION

METHODMETHOD

Lesson outcome:

Apply contradiction method to prove a

theorem.