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lecture13 - 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 Justify your answers Give an explanation • If P ~Q is false. What is the truth value of Q? Answer : Justify : Give a proof • If n is even, then n 2 is even. True or false. Answer : Justify : What are you expected to do?
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Lecture 13 3 Justify your answers Give an example • If n is odd, then mn is odd for all integers m. True or false. Answer : Justify : Show the working • Find the positive integer n < 7 with 3 10 ª n (mod 7) Answer : Justify : What are you expected to do?
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Lecture 13 4 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 • yes or no • truth values • numerical values
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Lecture 13 5 Justify a statement is true Universal statement Direct proof Proof by contrapositive Proof by contradiction Using cases Existential statement Constructive proof Non-constructive proof
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Lecture 13 6 Justify a statement is false Universal statement (To show its negation is true) • Constructive proof • Non-constructive proof Existential statement (To show its negation is true) • Direct proof • Proof by contrapositive • Proof by contradiction • Using cases
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Lecture 13 7 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) Ø 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
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Lecture 13 8 True or false To determine whether a statement is true or false universal statement existential statement
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