lecture13(complete) - MA1100 Lecture 13 Revision Lecture...

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1 MA1100 Lecture 13 Revision Lecture Justification Quantified Statements Proving Set Relations Using Hypothesis Without Loss of Generality Common mistakes Chartrand: chapters 1 - 5
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Lecture 13 2 Announcement ± Today: Collect back your HW2 ± Tomorrow: Online survey closed ± Thursday: Chat room with VT 7.30pm ± Friday: Mid-term test 12.00pm ± Next week : ² Tutorial 5 as usual ² (details later)
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Lecture 13 3 Justify your answers ± Give an explanation •I fP ~Q is false. What is the truth value of Q? Answer : True Justify : Explain why truth value of Q is true. ± Give a proof • If n is even, then n 2 is even. True or false. Answer : True Justify : Give a direct proof
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Lecture 13 4 Justify your answers ± Give an example • If n is odd, then mn is odd for all integers m. True or false. Answer : False Justify : Give counter-examples of n and m. ± Show the working • Find the positive integer n < 7 with 3 10 ª n (mod 7) Answer : 4 Justify : Show the working for deriving 3 10 ª 4 (mod 7) What are you expected to do? L Q 1
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Lecture 13 5 Justify your answers ± Depends on the type of question • about universal statement • about existential statement • about logical statements • about set relations •others ± Depends on your answer • true or false •yesorno •truthva lues
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Lecture 13 6 Justify a statement is true ± Universal statement •D i r e c t p r o o f • Proof by contrapositive • Proof by contradiction • Using cases ± Existential statement • Constructive proof (give an example) • Non-constructive proof (direct or indirect proof) indirect proof (with direct or indirect proof)
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Lecture 13 7 Justify a statement is false ± Universal statement (To show its negation is true) • Constructive proof (give a counter-example) • Non-constructive proof (direct or indirect proof) ± Existential statement (To show its negation is true) •D irectproo f • Proof by contrapositive
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Lecture 13 8 Counter-example To justify a universal statement is false ( " x) [P(x)] Give an example of x such that P(x) is false ( " x) [P(x) Q(x)] Give an example of x such that P(x) is true but Q(x) is false ( " x)( $ y) [P(x, y)] Give an example of x such that ( " y) [~ P(x, y)] is true brief explanation brief explanation proof ( " x) ( " y) [P(x, y)] Give examples of x, y
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Lecture 13 9 True or false To determine whether a statement is true or false universal statement existential statement construct examples violate statement don’t violate statement conclude statement false proceed to prove statement construct examples violate statement don’t violate statement proceed to disprove statement conclude statement true
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Lecture 13 10 Universal vs. existential ± Look for key words (quantifier) • Universal: (For) all, (for) every, (for) any
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This note was uploaded on 04/15/2010 for the course MATHS MA1101R taught by Professor Vt during the Spring '10 term at National University of Singapore.

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lecture13(complete) - MA1100 Lecture 13 Revision Lecture...

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