Discrete Mathematics - Discrete Math - Lecture Slides, Slides of Discrete Mathematics

Download Discrete Mathematics - Discrete Math - Lecture Slides and more Slides Discrete Mathematics in PDF only on Docsity! Introduction to Discrete Mathe ma tic s Docsity.com What did our brains evolve to do? Wha t we re our b ra ins “inte llige ntly de s igne d” to do? Wha t kind of me a t d id the Flying Spa ghe tti Mons te r put in our he a ds ? Docsity.com Our brains can per form only simple, c onc re te ta s ks And tha t’s how ma th s hould be a pp roa che d ! Dra w s imp le p ic ture s Try out s ma ll e xa mp le s of the p roble m: Wha t ha ppe ns for n=1? n=2? Subs titu te c onc re te va lue s for the va r ia b le s : x=0, x=100, … Docsity.com Exper t Novic e Docsity.com • The better the problem solver, the less brain activity is evident. The real masters show almost no brain activity! Simple and to the point Docsity.com Quick Test... Count the gre e n s qua re s … you will ha ve thre e s e c onds … Docsity.com Docsity.com How many were there? Docsity.com Hats with Consecutive Numbers Alic e Bob Alic e s ta r ts : … | X - Y | = 1 a nd X, Y > 0 I d on’t know wha t my numbe r is (round 1) Docsity.com Hats with Consecutive Numbers Alic e Bob Alic e s ta r ts : … | X - Y | = 1 a nd X, Y > 0 I don’t know wha t my numbe r is (round 2) Docsity.com Hats with Consecutive Numbers Alic e Bob Alic e s ta r ts : … | X - Y | = 1 a nd X, Y > 0 I d on’t know wha t my numbe r is (round 3) Docsity.com Hats with Consecutive Numbers Alic e Bob Alic e s ta r ts : … | X - Y | = 1 a nd X, Y > 0 I know wha t my numbe r is !!!!!!!! (round 251) Docsity.com Hats with Consecutive Numbers Alic e Bob Alic e s ta r ts : … | X - Y | = 1 a nd X, Y > 0 I know wha t my numbe r is !!!!!!!! (round 252) Docsity.com Question: What are Alice a nd Bob’s numbe rs ? Docsity.com If Alice knows in round 1, then X = 2 a nd Y = 1 If Alic e doe s not know in round 1, but Bob knows in round 2, the n X = 1 a nd Y = 2 If Bob d oe s not know in round 2, but Alic e knows in round 3, the n X = 3 a nd Y = 2 If Alic e doe s not know in round 3, but Bob knows in round 4, the n X = 2 a nd Y = 3 If Bob d oe s not know in round 250, but Alic e knows in round 251, the n X = 127 a nd Y = 126 : Docsity.com Magnets n ma gne ts a re c lumpe d Whe n the c lump is s p lit up in to n 1 a nd n 2 p ie c e s , the e ne rgy s pe nt is n 1n2 Wha t’s the le a s t a mount of e ne rgy ne c e s s a ry to s p lit the c lump into s ing le ma gne ts ? Docsity.com Exemplification: Try out a problem or solution on small examples. Look for the patterns. Docsity.com Four Magnets Ene rgy = 6 4 1 1 3 2 1 Docsity.com Every time you split a magnet from m other ma gne ts , you mus t s pe nd m units of e ne rgy Eve ry ma gne t mus t be s e pa ra te d from n- 1 othe r ma g ne ts Ene rgy = n(n-1)/2 Docsity.com • A volunteer, please Docsity.com Get The Problem Right! • Given any context you should double check that you read/heard it correctly! • You should be able to repeat the problem back to the source and have them agree that you understand the issue Docsity.com • Four guys want to cross a bridge that can only hold two people at one time. It is pitch dark and they only have one flashlight, so people must cross either alone or in pairs (bringing the flashlight). Their walking speeds allow them to cross in 1, 2, 5, and 10 minutes, respectively. Is it possible for them to all cross in 17 minutes? Docsity.com Intuitive, But False • “10 + 1 + 5 + 1+ 2 = 19, so the four guys just can’t cross in 17 minutes” • “Even if the fastest guy is the one to shuttle the others back and forth – you use at least 10 + 1 + 5 + 1 + 2 > 17 minutes” Docsity.com • If it is possible, there must be more than one guy doing the return trips: it must be that someone gets deposited on one side and comes back for the return trip later! Docsity.com • Suppose we leave 1 for a return trip later • • We start with 1 and X and then X returns Total time: Thus, we star t with 1,2 go over and 2 comes back…. 2X Docsity.com • 1 2 5 10 • Docsity.com • 1 2 5 10 • 5 10 2 1 Docsity.com • 1 2 5 10 • 5 10 2 5 10 2 1 1 Docsity.com • 1 2 5 10 • 5 10 2 5 10 2 1 1 Docsity.com • 1 2 5 10 • 5 10 2 5 10 • 2 • 1 2 2 1 1 1 5 10 5 10 Docsity.com • 1 2 5 10 • 5 10 2 5 10 • 2 • 1 2 2 1 1 1 5 10 5 10 Docsity.com • 1 2 5 10 • 5 10 2 5 10 • 2 • 1 2 2 1 1 1 5 10 5 10 1 2 5 10 Docsity.com Words To The Wise • Keep It Simple • Don’t Fool Yourself Docsity.com That really was a Microsoft question Why do you th ink tha t the y a s k s uc h que s tions , a s oppos e d to a s king for a p ie c e of c ode to do b ina ry s e a rc h? Docsity.com The future belongs to the computer scientist who has – Content: An up to date grasp of fundamental problems and solutions – Method: Principles and techniques to solve the vast array of unfamiliar problems that arise in a rapidly changing field Docsity.com Prover Ve r ifie r S ta te me nt1 S ta te me nt2 S ta te me ntn … The re is no s ound re a s on to go from Sta ta me nt1 to S ta te me nt2 Docsity.com Ver ifier The ve r ifie r is ve ry thorough, (he c a n c a tc h a ll your mis ta ke s ), but he will not s upp ly mis s ing d e ta ils of a p roof A va lid c ompla int on his pa r t is : I don’t unde rs ta nd The ve r ifie r is s imila r to a c ompute r running a p rogra m tha t you wrote ! Docsity.com Undefined term Synta x e rror Writing Proofs Is A Lot Like Writing Progra ms You ha ve to write the c orre c t s e que nc e of s ta te me nts to s a tis fy the ve r ifie r Infin ite Loop Output is not quite wha t wa s ne e de d Errors tha n c a n oc c ur with a p rogra m a nd with a p roof! Docsity.com Prover Ve r ifie r A s uc c e s s ful ma the ma tic ia n p la ys both role s in the ir he a d whe n writing a p roof Docsity.com Docsity.com Proof by Throwing in the Kitc he n S ink The a uthor wr ite s down e ve ry the ore m or re s ult known to ma nkind a nd the n a dds a fe w more jus t for good me a s ure Whe n que s tione d la te r, the a uthor c orre c tly obs e rve s tha t the p roof c onta ins a ll the ke y fa c ts ne e de d to a c tua lly p rove the re s ult Ve ry p opula r s tra te gy on e xa ms Be lieve d to re s ult in e xtra c re d it with s uffic ie nt whining 10 Docsity.com Proof by Cumbersome Notation Be s t done with a c c e s s to a t le a s t four a lpha be ts a nd s pe c ia l s ymbols . He lp s to s pe a k s e ve ra l fore ign la ngua ge s. 8 Like wr iting a program that ’s really hard to read because the var iable names are screwy Docsity.com Proof by Lengthiness An is s ue or two of a journa l de vote d to your p roof is us e ful. Works we ll in c omb ina tion with p roof by throwing in the kitc he n s ink a nd p roof by c umbe rs ome nota tion. 7 Like wr iting 10,000 lines of code to simply pr int “hello world” Docsity.com Proof by Switcharoo Conc lud ing tha t p is true whe n both p ⇒ q a nd q a re true 6 If (PRINT “X is pr ime” ) { PRIME(X); } Makes as much sense as: Docsity.com Not Cover ing All Cases Us ua l mis ta ke in induc tive p roofs : A p roof is g ive n for N = 1 (ba s e c a s e ), a nd a nothe r p roof is g ive n tha t, for a ny N > 2, if it is true for N, the n it is true for N+1 3 RECURSIVE(X) { if (X > 2) { return 2*RECURSIVE(X-1); } if (X = 1) { return 1; } } Like a program with this function: Docsity.com Incorrectly Using “By Definition” “By de finition, { a nb n | n > 0 } is not a re gula r la ngua ge ” 2 Like a program that assumes a procedure does something other than what it actually does Docsity.com Proof by OMGWTFBBQ 1/20 1 Docsity.com