Administrative Information. COS 341 Discrete Mathematics. Administrative Information. Discussion Sessions/Office Hours

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1 Admiistrative Iformatio COS 34 Discrete Mathematics Professor: Moses Charikar 35 CS buildig, Secretary: Mitra Kelly 33 CS Buildig, Teachig Assistats: Adriaa Karagiozova 44 CS Buildig, Reato Wereck 34 CS Buildig, Admiistrative Iformatio Required text: Ivitatio to Discrete Mathematics Jiri Matousek ad Jaroslav Nesetril Referece: Discrete Mathematics ad its Applicatios Keeth Rose All hadouts posted o web page Class mailig list Homeworks assiged every Wed., due i class ext Wed. Gradig: 9- homeworks (8%), take-home fial (%) Collaboratio policy Discussio Sessios/Office Hours Discussio Sessios i additio to office hours (perhaps) Times aouced ext week o basis of studet choices My office hours for ext week: Tue, :-4: pm 3 4

2 What is Discrete Mathematics? Mathematics dealig with fiite sets Topics: coutig, combiatorics, graph theory, probability Goals: Develop mathematical maturity Foudatio for advaced courses i Computer Sciece Flavor of questios: How may valid passwords o a computer system? What is the probability of wiig a lottery? What is the shortest path betwee two cities? Toy problems as illustratios Puzzle: Three houses, three wells: Ca we coect each house to each well by pathways so that o two pathways cross? Real world problem: VLSI: Give placemet of compoets of circuit o a board, is it possible to coect them alog a board so that o two wires cross? 5 6 Proof techiques p ( ) = Claim: Ν, p( ) is prime 7 8

3 Evidece p() = 4 prime p() = 43 prime p() = 47 prime p(3) = 53 prime p () = 46 prime looks promisig! p (39) = 6 prime must be true! Oly prime umbers? Ν, p = + + ( ) 4is prime Caot be a coicidece Hypothesis must be true! But it is NOT! p (4) = 68 is NOT PRIME 9 Euler s cojecture (769): a + b + c = d has o solutio for abcd,,, positive iteges r Couterexample: 8 years later by Noam Elkies: = 448 Example courtesy: Prof. Albert R. Meyer s lecture slides for MIT course 6.4, Fall Hypothesis: 33 ( x + y ) = z has o positive iteger solutio False! But smallest couterexample has more tha digits! Example courtesy: Prof. Albert R. Meyer s lecture slides for MIT course 6.4, Fall

4 Mathematical Iductio Fid a geeral formula for i = + = = = 5 i + =? 3 Mathematical Iductio + =. (Basis step) i + = holds for = i. (Iductive step) Suppose formula holds for = (iductive hypothesis) We prove that it also holds for = + + i i = + = + ( ) + = basis step iductive step Show that ay chessboard with oe square removed ca be tiled usig L-shaped pieces, each coverig three squares. 5 6

5 Template for proof by Iductio Strog Iductio Prove that P() is true for all positive itegers BASIS STEP: Show that P() is true INDUCTIVE STEP: Show that P(k) P(k+) Prove that P() is true for all positive itegers BASIS STEP: Show that P() is true INDUCTIVE STEP: Show that [ P() P() P( k)] P( k+ ) 7 8 Strog Iductio example Show that if is a iteger greater tha, ca be writte dow as a product of primes P(): ca be writte dow as a product of primes Basis Step: P() is true, = Iductive Step: Assume P(j) is true for all j k Need to show that P(k+) is true Case : k+ is prime, P(k+) is true Case : k+ = a b By iductio hypothesis, both a ad b ca be writte as a product of primes. 9 Proof by cotradictio Prove that is irratioal m Suppose = gcd( m, ) = m = m is eve m = k 4k = = k is eve Cotradictio!