Lecture_12 - MA1100 Lecture 12 Mathematical Induction Using...

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1 MA1100 Lecture 12 Mathematical Induction Using PMI on other universal sets Strong PMI Variations of MI
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MA1100 Lecture 12 2 Other Universal Sets Can we use PMI to prove statements of the form ( " x œ Q + ) P(x) ? ( " x œ R + ) P(x) ? Possible. No. ( " x œ Z ) P(x) ? Possible.
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MA1100 Lecture 12 3 Other Universal Sets ( " n œ Z )P(n) 1. P(0) is true 2. For all k ¥ 0, if P(k) is true, then P(k+1) is true 3. For all k § 0, if P(k) is true, then P(k - 1) is true Then P(n) is true for all n œ Z P(0) Ø P(1) Ø P(2) Ø Ø P(n)… …P(-n) P(-2) P(-1) P(0)
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MA1100 Lecture 12 4 Other Universal Sets ( " q œ Q + )P(q) 1. P( 1 ) is true 2. For all k œ Z + ,ifP( k ) is true, then P( k+1 ) is true Then P(m/n) is true for all m/n œ Q + + where m, n m q n = Z Fix m œ Z + Then P(n) is true for all n œ Z + 3. P(m/ 1 ) is true (from 1 and 2 above) 4. For all k œ Z + ,ifP(m/ k ) is true, then P(m/ (k+1) ) is true
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MA1100 Lecture 12 5 Other Universal Sets ( " q œ Q + )P(q) + where m, n m q n = Z P(1) Ø P(2) Ø P(3) Ø Ø P(m) … ) n 1 ( P ) 3 1 ( P ) 2 1 ( P # ) n 2 ( P ) 3 2 ( P ) 2 2 ( P # ) n 3 ( P ) 3 3 ( P ) 2 3 ( P # ) n m ( P ) 3 m ( P ) 2 m ( P #
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MA1100 Lecture 12 6 Binary Sum Proposition Every positive integer can be written as the sum of distinct power of 2 . 4 = 2 2 10 = 2 3 + 2 1 45 = 2 5 + 2 3 + 2 2 + 2 0 Example Proof by Mathematical Induction Base case : prove P(1) is true P(n): n is a sum of distinct power of 2 . Since 1 = 2 0 , So P(1) is true. Binary Sum
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MA1100 Lecture 12 7 Binary Sum Inductive step : P(k) Ø P(k+1) for all k ¥ 1 hypothesis P(k): want to get P(k+1): Is knowing P(k) enough to prove P(k+1) ? P(n): n is a sum of distinct power of 2 . k is a sum of distinct power of 2 k + 1 is a sum of distinct power of 2 Let’s first try the ordinary PMI : k = sum of distinct power of 2 k + 1 = (sum of distinct power of 2) + 2 0
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Lecture 12 8 Strong PMI Let P(n) be a predicate such that: 1. P(1) is true 2. For all k œ Z + , if P(1), P(2), …, P(k) are true, then P(k+1) is true. Then P(n) is true for all n
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Lecture_12 - MA1100 Lecture 12 Mathematical Induction Using...

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