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Unformatted text preview: 111ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiICS141: Discrete Mathematics for Computer Science IDepartment of Information and Computer SciencesUniversity of HawaiiStephen Y. Itoga112ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiLecture 11Chapter 3. The Fundamentals3.1 Algorithms113ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiSome material in these slides were taken/adapted from the slides made by Prof. Michael P. Frank and Prof. Jonathan L. Gross which are provided through the publisher of Discrete Mathematics and Its Applications written by Kenneth H. Rosen.Some material is from Prof. Baek114ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiSummation NotationGiven a sequence {an}, an integer lower bound (orlimit) j, and an integer upper bound kj, then the summation of {an} from ajto akis written and defined as follows:kjjkjiiaaaa+++=+=...1=====kjllkjmmkjiiaaa115ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiNested SummationsThese have the meaning youd expect.Note issues of free vs. bound variables, just like in quantified expressions, integrals, etc.( 2960106)4321(666321414141413141314131==+++===++=== =========iiiijijijiiijiijij116ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiSome Shortcut ExpressionsGeometric sequenceEulers trickQuadratic seriesCubic series117ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiUsing the ShortcutsExample: Evaluate=100502kk.925,297425,40350,33869950496201101100491210012100502100502491210012====+=======kkkkkkkkkkkkUse series splitting.Solve for desiredsummation.Apply quadraticseries rule.Evaluate.118ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiSummations: ConclusionYou need to know:How to read, write & evaluate summation expressions like:Summation manipulation laws we covered.Shortcut closedform formulas, & how to use them.Xxxf)()()(xPxf=kjiia=jiia119ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiThe sets Aand Bhave the same cardinalityif and only if there is a onetoone correspondence from Ato B. A set that is either finite or has the same cardinality as the set of positive integers is called countable. A set that is not countable is called uncountable. Example: Show that the set of odd positive integers is a countable set.Cardinality (I)A onetoone correspondence between Z+and the set of odd positive integers.Consider the functionf(n) = 2n 1from Z+to the set of odd positive integers1110ICS 141: Discrete Mathematics I (Spr 2008)University of Hawaiia1, a2,,an, expresses onetoone correspondence f: Z+S where a1=f(1), a2=f(2),, an=f(n),Example: Show that the set of positive rational numbers is countable (see figure)...
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