Wrap of Number Theory & Midterm Review. Recall: Fundamental Theorem of Arithmetic

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1 Wrap of Number Theory & Midterm Review F Primes, GCD, ad LCM (Sectio 3.5 i text) F Midterm Review Sectios.-.7 Propositioal logic Predicate logic Rules of iferece ad proofs Sectios.-.3 Sets ad Set operatios Fuctios Sectios Itegers, div, mod, cogruece, applicatios Primes ad their properties Recall: Fudametal Theorem of Arithmetic

2 Fudametal Theorem of Arithmetic F FTA Theorem. Z + where >, is a prime or a product of primes i odecreasig order. (Proof i a later sectio) F I other words, primes are the buildig blocks of itegers F FTA examples: 50 = x 5 x 5 = 5 7 = x x x 3 x 3 = = 5 3 Testig whether a umber is prime F Naïve algorithm for primality testig: Iput : For a =,, -:Test whether a. If o a divides, the prime. F Is there a better (faster) algorithm? Do we eed to test all the umbers from to -? 4

3 Testig whether a umber is prime F Thm: composite has a prime factor Proof: composite a (<a<) = ab for some iteger b >. Suppose a > ad b >. The ab > i.e., ab >. This cotradicts ab =. Therefore, a or b. If a or b is prime, we are doe. Otherwise, by FTA, a is product of prime factors < a ad b is product of prime factors < b. Therefore, has a prime factor. QED. F Corollary: If does ot have a prime factor, the is prime 5 Algorithm for Primality 6

4 Algorithms for Primality ad Prime Factorizatio F Algorithm for Primality: Give, test whether ay prime from to divides. If oe does, the is prime. Example: Is 3 a prime? Test, 3, 5, 7,, 3, 7 3 Noe divides 3, therefore 3 is a prime. (Note: oly tested 7 umbers istead of the 309 umbers i the aïve algorithm!) F Algorithm for prime factorizatio of : Fid prime factors p p / p, p /( p )..., 3 p F Example: Fid prime factorizatio of Test, 3, , so p = 3; Next, 89/3 = Test, 3, 3 73, so p = 3; Next, 73/3 = 9 9 Test, 3, 5, 7 7 9, so p 3 = 7; Next, 9/7 = 3 (a prime) Therefore, 89 = Ai t primal euff for me, mate! 8

5 How may primes are there? F Euclid s theorem (circa 300 BC): There are ifiitely may primes. Proof by cotradictio: See text. Corollary: For ay positive iteger, there is always a prime greater tha. F How may primes? Let P() = umber of primes. Prime Number Theorem: P() is approximately / l as grows without boud. Cor.: Probability that a radom positive it. is prime = (/ l )/ = / l P() / l 9 Greatest Commo Divisor (GCD) F Example: Positive divisors of 6 =,, 4, 8, 6 Positive divisors of 4 =,, 3, 4, 6, 8, Greatest Commo Divisor gcd(6,4) = 8 F For ay ozero a,b Z, gcd(a,b) = largest iteger d such that d a ad d b gcd(0,5) = 5, gcd(7,5) = a, b are relatively prime iff gcd(a,b) =. E.g., 7 ad 5. F Computig gcd(a,b): Use prime factorizatio of a, b a a p a gcd( a, b) a mi( a, b ) 3 0 E.g , 7 3, gcd(60,7) 3 5, b p b p b mi( a, b ) b ( a, b ca be 0) i mi( a, b ) i 0

6 Least Commo Multiple (LCM) F Example: Multiples of 6 = 6,, 8, 4, 30, Multiples of 8 = 8, 6, 4, 3, Least Commo Multiple lcm(6,8) = 4 F For ay a,b Z +, lcm(a,b) = smallest c Z + such that a c ad b c. lcm(4,6) =, lcm(5,0) = 0, lcm(5,) = 55 F Computig lcm(a,b): Use prime factorizatio of a, b a a p a lcm( a, b) a max( a, b ), b b b max( a, b ) b 3 3 E.g. 6 3, 8,lcm(6,8) 3 4 F Theorem: gcd(a,b) lcm(a,b)=ab p p ( a, b ca be 0) i max( a, b ) i Midterm Review: Chapter (Sectios.-.7) F Propositioal Logic Propositios, logical operators,,,,,, truth tables for operators, precedece of logical operators Compoud propositios, truth tables for compoud propositios Coverse, cotrapositive, ad iverse of p q Covertig from/to Eglish ad propositioal logic F Propositioal Equivaleces Tautology versus cotradictio Logical equivalece p q Tables of logical equivaleces (tables 6, 7, 8 i text) De Morga s laws Showig two compoud propositios are logically equivalet via (a) truth table method ad (b) via equivaleces i tables 6, 7, 8.

7 Predicate Logic F Predicates ad Quatifiers Predicates, variables, ad domai of each variable Uiversal ad existetial quatifiers ad (uiqueess!) Truth value of a quatifier statemet Logical equivalece of two quatified statemets Negatio ad De Morga s laws for quatifiers Traslatig to/from Eglish F Nested Quatifiers Traslatig to/from Eglish, egatig ested quatifiers 3 Rules of Iferece Modus poes Modus borus Modus tolles 4

8 Rules of Iferece F Rule of iferece = valid argumet form. Table (p. 66). Modus poes: [p (p q)] q Modus tolles: [(p q) q] p Hypothetical Syllogism: [(p q) (q r)] (p r) Disjuctive Syllogism: : [(p q) p] q Additio, Simplificatio, Cojuctio Resolutio: [(p q) ( p r)] (q r) F Usig rules of iferece to prove statemets from premises F Rules of iferece for quatified statemets: istatiatio ad geeralizatio 5 Proofs ad Proof Methods F Direct proof of p q: Assume p is true; show q is true. F Proof of p q by cotrapositio: Assume q ad show p. F Vacuous ad Trivial Proofs of p q F Proof by cotradictio of a statemet p: Assume p is ot true ad show this leads to a cotradictio (r r). F Proofs of equivalece for p q: Show p q ad q F Proof by cases ad Existece proofs 6

9 Chapter : Sets ad Operatios (Sectios.-.) F Sets Set builder otatio, set equality, Ve diagrams Sets Z, Z +, R, Q, N,, sigleto sets Subset ad proper subset Cardiality, fiite ad ifiite sets, Power set Tuples, Cartesia product, truth set of a predicate F Set operatios,, differece, complemet Set idetities (similar to logical equivaleces) Provig two sets are equal: Two methods Show each set is a subset of the other, OR Use logical equivaleces F Bit strig represetatio of sets ad bitwise operatios 7 Chapter : Fuctios (Sectio.3) F Defiitio of a fuctio Domai, co-domai, rage, image, preimage - ad oto fuctios, bijectios Kow defiitios ad how to show -, oto, or bijectio Iverse of a fuctio ad compositio of fuctios floor ad ceilig fuctios Kow defiitios ad how to compute 8

10 Chapter 3: Itegers ad Divisio (Sectio 3.4) F Divisio Kow defiitios of a b, factor, multiple Prove idetities ivolve Divisio algorithm Kow the statemet, div, mod F Modular arithmetic Kow defiitio ad theorems a b (mod m) iff m (a-b) iff a mod m = b mod m iff a = b + km 9 Applicatios of Modular Arithmetic F Hashig Hashig fuctio Collisio 0

11 Applicatios of Modular Arithmetic Pseudoradom umbers usig liear cogruetial geerator Applicatios of Modular Arithmetic Cryptology F Caeser s cipher F Shift cipher F Ecryptio F Decryptio

12 Chapter 3: Primes ad GCD (Sectio 3.5) F Primes Defiitio, Fudametal Theorem of Arithmetic (FTA) Algorithms for testig primality ad prime factorizatio Euclid s ifiitude of primes theorem Prime umber theorem: Number of primes ot exceedig is approximately / l as grows without boud F GCD ad LCM Defiitio of gcd ad lcm, defiitio of relatively prime Fidig gcd ad lcm through prime factorizatios (usig mi/max of expoets) 3 Good luck o the midterm! F You ca brig oe 8 /'' x '' review sheet (double-sided ok, hadwritte or typed but o magifyig aids please!). F Calculators okay to use but wo t really eed it. Do t sweat it! Go through the homeworks, lecture otes, ad examples i the text Do the practice midterm o the website ad avoid beig surprised! 4