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Analysis 1.5

# Analysis 1.5 - ” • The notation was popularized in the...

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Last Updated: 10/01/12 5:45 AM CSE 2011 Prof. J. Elder - 21 - Which are more alike? Quadratic 1000n 2 3n 2 2n 3

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Last Updated: 10/01/12 5:45 AM CSE 2011 Prof. J. Elder - 22 - properties of logarithms: log b (xy) = log b x + log b y log b (x/y) = log b x - log b y log b x a = alog b x log b a = log x a/log x b properties of exponentials : a (b+c) = a b a c a bc = (a b ) c a b /a c = a (b-c) b = a log a b b c = a c*log a b Summations Logarithms and Exponents Existential and universal operators Proof techniques Some Math to Review ! g " b Loves( b , g ) " g ! b Loves( b , g ) existential and universal operators
Last Updated: 10/01/12 5:45 AM CSE 2011 Prof. J. Elder - 23 - ! g , " b , loves ( b , g ) ! g , " b , loves ( b , g ) Understand Quantifiers!!! One girl Could be a separate girl for each boy. Sam Mary Bob Beth John Marilyn Monro Fred Ann Sam Mary Bob Beth John Marilyn Monro Fred Ann

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Last Updated: 10/01/12 5:45 AM CSE 2011 Prof. J. Elder - 24 - Asymptotic Notation The notation was first introduced by number theorist Paul Bachmann in 1894, in the second volume of his book Analytische Zahlentheorie
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Unformatted text preview: ”). • The notation was popularized in the work of number theorist Edmund Landau ; hence it is sometimes called a Landau symbol. • It was popularized in computer science by Donald Knuth , who (re)introduced the related Omega and Theta notations. • Knuth also noted that the (then obscure) Omega notation had been introduced by Hardy and Littlewood under a slightly different meaning, and proposed the current definition. Source: Wikipedia ( ! , " , # and all of that) Last Updated: 10/01/12 5:45 AM CSE 2011 Prof. J. Elder - 25 - Big-Oh Notation • Given functions f ( n ) and g ( n ) , we say that f ( n ) is O ( g ( n )) if there are positive constants c and n such that f ( n ) ≤ cg ( n ) for n ≥ n 0 • Example: 2 n + 10 is O ( n ) – 2 n + 10 ≤ cn – ( c ± 2) n ≥ 10 – n ≥ 10 / ( c ± 2) – Pick c = 3 and n 0 = 10 1 10 100 1,000 10,000 1 10 100 1,000 n 3n 2n+10 n...
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