Chapter 4 Proof Techniques.pdf

# Chapter 4 Proof Techniques.pdf - BUM1233 APPLICATIONS...

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BUM1233 D ISCRETE M ATHEMATICS & A PPLICATIONS C HAPTER 4: P ROOF T ECHNIQUES Mohd Sham Mohamad ([email protected]) Chapter 4: Proof Techniques

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4.1 Introduction to Proof Techniques 4.2 Direct Method 4.3 Indirect Method 4.4 Contradiction Method 4.5 Mathematical Induction 4.6 Strong Induction and Well-Ordering CHAPTER 4: PROOF TECHNIQUES Chapter 4: Proof Techniques
Lesson Outcome: Understand some terms and terminologies in proving technique 4.1 INTRODUCTION TO PROOF TECHNIQUES Chapter 4: Proof Techniques

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TERM DESCRIPTION Theorem A theorem is a statement that can be shown to be true. Theorems can also be referred as facts or results. Proof 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. Proposition Less important theorems sometimes are called propositions. Lemma A lemma is a less important theorem that is helpful in the proof of other result. Corollary A corollary is a theorem that can be established directly from a theorem that has been proved. Conjecture A corollary is a statement that is being proposed to be a true statement. TERMS Chapter 4: Proof Techniques
ODD & EVEN n is ODD if: n is EVEN if: 2 1 k n k     2 k n k     7 is odd since there exist 3 such that 7=2(3)+1 100 is even since there exist 50 such that 100=2(50) Chapter 4: Proof Techniques

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METHOD OF PROOF To prove the theorem given in implication (if p then q , ), we have three methods of proof: p q 1 Direct Method 2 Indirect Method Contrapositive Method 3 Contradiction Method Chapter 4: Proof Techniques
Lesson Outcome: Apply direct method to prove a theorem 4.2 DIRECT METHOD Chapter 4: Proof Techniques

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ASSUME & SHOWS Assume the results (theorem etc.) are given in implication proposition logic (if p then q , ). Direct method needs us to assume p is true and shows that q is true. p q Example: Prove that if x is even, then is even 2 x Chapter 4: Proof Techniques
Proof: Let : even p x 2 : even q x . Assume that x is even is true. Need to prove that 2 x is even. Since x is even, 2 , x k k . Then 2 2 2 2 (2 ) 4 2(2 ) 2 ( 2 ) x k k k m m k Therefore 2 x is even. Example 1 Prove that if x is even, then is even 2 x Chapter 4: Proof Techniques

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Proof: Let : , odd p x y and : even q x y . Assume that and x y are odd is true. Need to prove that x y is even. Since x and y are odd, then 2 1, 2 1 x k y l where , k l . Thus,   2 1 2 1 2 1 2 x y k l k l m          where 1 m k l     . Therefore x y is even. ■ Example 2 Prove that if x and y are odd, then x+y is even Chapter 4: Proof Techniques
Example 3 Prove that if a and b are both perfect squares integer, then ab is also a perfect square integer.

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