CHAPTER 7: PROOF TECHNIQUES

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CHAPTER 7 PROOF TECHNIQUES

• Fundamental Discrete Structure
• BCT 1073

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CONTENT

• 7.1 Introduction
• 7.2 Direct method
• 7.3 Indirect method
• 7.4 Contradiction method
• 7.5 Mathematical Induction

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OBJECTIVES

• At the end of this chapter you should be able to
• Apply direct method, indirect method,
  contradiction, and mathematical induction to
  prove a theorem.

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7.1 INTRODUCTION
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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.

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Some Terminology

• Less important theorems sometimes are called
  propositions.
• A lemma is a less important theorem that is
  helpful in the proof of other result.
• A corollary is a theorem that can be established
  directly from a theorem that has been proved.
• A conjecture is a statement that is being
  proposed to be a true statement.

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Applications

• It is used to prove results about the
• complexity of algorithms
• theorems about graphs and trees
• wide range of identities and inequalities.

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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

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Methods of Proof

• To prove the theorem, we have four methods of
  proof
• (a) Direct method
• (b) Indirect method
• (c) Contradiction method
• (d) Mathematical Induction

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Useful Definitions
The integer n is even if there exists an integer
k such that n 2k.
The integer n is odd if there exists an integer k
such that n 2k 1.

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

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7.2 DIRECT METHOD
Lesson outcome Apply direct method to prove a
theorem.
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DIRECT METHOD

• A direct proof shows that a conditional statement
  p?q is true by showing that
• if p is true, than q must also be true.
• We assume that p is true and use axioms,
  definitions, and previously proven theorems,
  together with rules of inference, to show that q
  must also be true.

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Example 7.2.1

• Give a direct proof of the theorem If n is an
  odd
• integer, then n2 is odd.

SOLUTIONS
Let n is an odd integer
We want to show that n2 is an odd integer.
Therefore, n2 is an odd integer.
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Example 7.2.2

• 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 b2).

SOLUTIONS
Let m and n are both perfect squares
We want to show that mn is perfect square.
Therefore, mn is perfect square.
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EXERCISE 7.2

• Use a direct method of proof to show that if x
  and y are odd integer then xy is also odd.
• Prove p ? q for the following p and q.
• p number can be divided by 6
• q number can be divided by 3.

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7.3 INDIRECT METHOD
Lesson outcome Apply indirect method to prove a
theorem.
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INDIRECT METHOD

• Is known as proof by contraposition.
• Let p ? q ,
• converse q ? p
• contrapositive q ? p
• inverse p ? q

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Example 7.3.1

• Give an indirect proof of the theorem If 3n2 is
  odd, then n is
• odd. (n is an integer)

SOLUTIONS
Let p 3n2 is odd, q n is odd Contrapositive
q ? p n is
even ? 3n2 is even
Let n is an even integer
We want to show that 3n2 is even.
Therefore, 3n2 is even.
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EXERCISE 7.3

• Use an indirect method of proof to show that if
  xy is odd integer, then x and y are also odd
  integers.
• Show that if n is an integer and n35 is odd,
  then n is even using a proof by contraposition.

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7.4 CONTRADICTION METHOD
Lesson outcome Apply contradiction method to
prove a theorem.
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CONTRADICTION METHOD

• Prove p ? q is true.
• Assume p and q are true and show that q must
  also be true.

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Example 7.4.1

• Give a proof by contradiction of the theorem If
  3n2 is odd,
• then n is odd.

Let p 3n2 is odd, q n is odd Contradiction
Assume p and q is true.
3n2 is odd and n is even

SOLUTIONS
n is even
Contradiction
3n2 is even.
Therefore, if 3n2 is odd, then n must be odd.
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Example 7.4.2

• Give a proof by contradiction of If x x x,
  then x 0.

SOLUTIONS
Let p x x x, q x 0 Contradiction
Assume p and q is true. x x x and x ?
0
Contradiction
Therefore, if x x x, then x must zero.
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EXERCISE 7.4

• Use a contradiction method of proof to show that
  if x and y are odd integer then xy is also odd.
• Show that if n is an integer and n35 is odd,
  then n is even using a proof by contradiction.

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7.5 MATHEMATICAL INDUCTION
Lesson outcome Apply mathematical induction to
prove a theorem.
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Motivation

• Suppose we have a ladder and we want to know
  whether we can reach every step on this ladder.
• We know 2 things
• (a) We can reach the first rung of the
  ladder.
• (b) If we can reach a particular rung of the
  ladder, then we can reach the next rung.

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MATHEMATICAL INDUCTION

• Objective To proof that the statement P(n) is
  true for each integer n n0
• Steps
• 1. Prove that the statement is true for n n0.
• 2. Assume that the statement is true for n k.
• 3. Prove that the statement is true for n k
  1.
• 4. Conclusion Therefore, the statement is true
  for each integer n n0

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How to Remember?

• Thinking of the ladder and the rules for reaching
  steps can help you remember how mathematical
  induction works.
• Another way
• (a) A line of people person 1, person 2, and so
  on. A secret is told to person 1, and each person
  tells the secret to the next person in line, if
  the former person hear it.
• (b) row of domino

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Example 7.5.1

• Use mathematical induction to show that
• for any positive integer n.

SOLUTIONS
Let
1. Prove that the statement is true for n n01
which is clearly true.
2. Assume that the statement is true for n k.
is true.
Let
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Example 7.5.1 (cont)

3. Prove that the statement is true for n k 1.



We want to prove that
4. Conclusion Therefore, the statement is true
for each integer n n0
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EXERCISE 7.5

• Show by mathematical induction that, for all n1,
  
• Show by mathematical induction that, for all n1,
  

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Thank You