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

Info icon This preview shows pages 1–12. Sign up to view the full content.

View Full Document Right Arrow Icon
BUM1233 D ISCRETE M ATHEMATICS & A PPLICATIONS C HAPTER 4: P ROOF T ECHNIQUES Mohd Sham Mohamad ([email protected]) Chapter 4: Proof Techniques
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 2
Lesson Outcome: Understand some terms and terminologies in proving technique 4.1 INTRODUCTION TO PROOF TECHNIQUES Chapter 4: Proof Techniques
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 4
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
Image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 6
Lesson Outcome: Apply direct method to prove a theorem 4.2 DIRECT METHOD Chapter 4: Proof Techniques
Image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 8
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
Image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 10
Example 3 Prove that if a and b are both perfect squares integer, then ab is also a perfect square integer.
Image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 12
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern