MA112-Lecture_Notes-OH-L05

MA112-Lecture_Notes-OH-L05 - MA112 Discrete Mathematics...

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MA112 Discrete Mathematics Dr. Omar Hamdy Summer 2010 1 MA112 Discrete Mathematics Omar Hamdy omar.hamdy@gmail.com Lecture Notes 5
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MA112 Discrete Mathematics Dr. Omar Hamdy Summer 2010 2 Review: Rules of Inference • Rules of inference means to build logically valid inference patterns or arguments. • Modus Ponens: – Rule of inference: p p q q – Read as: given p is true and the implication p q is true, then q is true.
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MA112 Discrete Mathematics Dr. Omar Hamdy Summer 2010 3 • Typically a theorem looks like this: (p 1 p 2 p 3 p 4 p n ) q Premises or Hypotheses Conclusion • To prove a theorem, we establish that the conclusion follows from the premises
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MA112 Discrete Mathematics Dr. Omar Hamdy Summer 2010 4 Type of Proofs p q is proved by: • showing that if p is true then q follows. (Direct Proof) • proving its contrapositive ¬q ¬p. (Indirect Proof) • proving its contradiction (p ¬q) is false (Contradiction Proof) • showing input p is always false (Vacuous Proof) • Showing output q is always true (Trivial Proof)
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MA112 Discrete Mathematics Dr. Omar Hamdy Summer 2010 5 Proof by Cases • Sometimes the hypotheses can be represented as p 1 , p 2 , …, p n , and we need to prove the conditional statement: • p 1 p 2 p n q • We can restructure the conditional statement to be: • (p 1 q) (p 2 q) (p n q) • Does that make sense? Can you prove it?
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MA112 Discrete Mathematics Dr. Omar Hamdy Summer 2010 6 Proof by Cases • p 1 p 2 p n q – ¬ (p 1 p 2 p n ) q – (¬ p 1 ¬p 2 ¬p n ) q – (¬ p 1 q) (¬ p 2 q) (¬ p n q) – (p 1 q) (p 2 q) (p n q) • If number of hypotheses (cases) is small, then we can derive the final proof by proving that each case is true.
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MA112 Discrete Mathematics Dr. Omar Hamdy Summer 2010 7 Proof by Cases • Show that |x||y|=|xy| • Recall |a| = a if a 0 and |a| = -a if a < 0 • We have 4 cases: – x 0: |x||y|= xy = |xy| – x 0: Similar to case ii • All cases proved.
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MA112 Discrete Mathematics Dr. Omar Hamdy Summer 2010 8 Proof by Cases • Can you make an observation about the final digit of the square of an integer and prove your observation? • 0 , 1 , 4 , 9 , 1 6 , 2 5 , 3 6 , 4 9 , 6 4 , 9 1 , 10 0 , 12 1 , 14 4 , 16 9 • Observation: last digit is either 0,1,4,5, 6 or 9 • To prove it, we first assume an integer n
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MA112 Discrete Mathematics Dr. Omar Hamdy Summer 2010 9 Proof by Cases • We can identify two cases for n: |n| < 10 and |n| 10. • For |n| < 10, we can prove by cases that the
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MA112-Lecture_Notes-OH-L05 - MA112 Discrete Mathematics...

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