CSE20_Lect15_sol.pdf

# CSE20_Lect15_sol.pdf - CSE 20 DISCRETE MATH Yeonkyung Kim...

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CSE 20 DISCRETE MATH Yeonkyung Kim Clicker frequency: AB Thursday, May 14, 15

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Induction Prove P(n) is true for n>=a. Base case : show that P(a) is true. Inductive step : 2 “for n>a, IF P(n-1) is true, then P(n) is true” n=a+1: P(a) ->P(a+1) n=a+2: P(a+1) ->P(a+2) n=a+3: P(a+2) ->P(a+3) Modus ponens n=a+4: P(a+3) ->P(a+4) n=a+5: P(a+4) ->P(a+5) n=a+6: P(a+5) ->P(a+6) P(a) is true + P(a+1) is true + P(a+3) is true + P(a+4) is true + P(a+2) is true + P(a+5) is true + P(a+1) is true P(a+2) is true P(a+3) is true P(a+4) is true P(a+5) is true P(a+6) is true Thursday, May 14, 15
Todays topics Proof by induction: Arithmetic sequence Geometric sequence Induction with sets Strong induction Section 3.6 in Jenkyns, Stephenson Thursday, May 14, 15

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Sequence Sequence : an ordered collection of objects(numbers) 2, 5, 6, 1, 7, 4, ... It may be finite/infinite Order matters 1,2,3,4,5... 1,4,2,5,3,... repetitions matter 1,2,3,4,5... 1,1,2,3,4,5 .... Different! Different! a 1 , a 2 , a 3 , a 4 , a 5 , ... a n : n th term Thursday, May 14, 15
Arithmetic sequence Sequence of numbers such that the difference between the consecutive terms is constant. d: the common difference Which of the following is NOT an arithmetic sequence? A:2,4,6,8,10... B:3,7,11,15,19,23,... C:2,3,4,5,6,... D:4,6,7,9,10,... E:5,7,9,11,13,15,... a n = a n - 1 + d common difference? Thursday, May 14, 15

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Arithmetic sequence Theorem: The general term Base case: n=?
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