mathematics for computer science book

... ιντεγερκ 2 ⇒ ν = κ , ωηιχη ισεϖεν 1/25/12 More slowly … • Thm For any integer n, n2 is odd if and only if n is odd • To prove a statement of the form “P iff Q,” two separate proofs are needed: – If P ... the assertions together are abbreviated “P iff Q” or “P⇔Q” or “P ≡Q” 1/25/12 More slowly … • Thm For any integer n, n2 is odd if and only if n is odd () “If n2 is odd then n is odd” is equivalent ... (“contrapositive”) which is the same as “if n is even then n2 is even” (since n is an integer) … then n=2k for some k and n2=4k2, which is even 1/25/12 Contrapositive and converse • The contrapositive of

Ngày tải lên: 22/03/2019, 10:35

... bottom pancake 03/22/19 Why does this take 2n-3 flips? • For n≥2, let P(n) := “n pancakes can be sorted using 2n-3 flips” • Suppose this is false for some n • Let C = {n: P(n) is false} • C has a least ... to “lowest terms”  The set of factors of a positive integer is nonempty 03/22/19 To prove P(n) for every nonnegative n: • Let C = {n: P(n) is false} (the set of “counterexamples”) • Assume C ... by WOP Call it m • So m pancakes cannot be sorted using 2m-3 flips and m is the smallest number for which that is the case 03/22/19 Why does this take 2n-3 flips? • m≠2 since one flip sorts pancakes

Ngày tải lên: 22/03/2019, 10:37

... Logic and computers 2/6/12 Binary Arithmetic Only two digits: the bits and (Think: = F, = T) +0 -0 2/6/12 +1 -1 +0 -1 +1 -10 Logic and Computers  A half adder:  Two ... Simpler formulas turn into circuits that use less hardware! • E.g p ⋁ q ⋁ (p⋀q) is equivalent to p ⋁ q but would use more logic gates • But the P=NP? question means that it may be hard to simplify formulas ... possible – Any tautology is equivalent to p ⋁ ¬p so if we could easily simplify formulas we could easily determine whether a formula is a tautology 2/6/12

Ngày tải lên: 22/03/2019, 10:41

... or “A is contained in B” (∀x) (x∈A ⇒ x∈B) N ⊆ Z, {7} ⊆ {7, “Sunday”, π} ∅ ⊆ A for any set A (∀x) (x∈∅ ⇒ x∈A) A ⊆ A for any set A To be clear that A ⊆ B but A ≠ B, write A ⊊ B “Proper subset” (I ... Sets 2/10/12 What is a Set? • Informally, a collection of objects, determined by its members, treated as a single mathematical object

Ngày tải lên: 22/03/2019, 10:45

... → R f(x,y) = x/y Defined for all pairs (x,y) except when y=0! 2/13/12 A Function that is “Partial,” Not Total domain f R×R codomain R f: R ×R → R f(x,y) = x/y Defined for all pairs (x,y) except ... “Size” For finite sets, a bijection exists iff A and B have the same number of elements domain A 2/13/12 f codomain B 10 Cardinality or “Size” Use the same as a definition of “same size” for infinite

Ngày tải lên: 22/03/2019, 10:47

... contradiction: suppose Pf f:A↔P(A) is a bijection Let W::= {a  A|a f(a)}, so for any a, a W iff a f(a) f is a bijection, so W=f(a0), for some a0 A (∀a) a f(a0) iff a f(a ) 2/22/12 12 There is no bijection ... contradiction: suppose Pf f:A↔P(A) is a bijection Let W::= {a  A|a f(a)}, so for any a, a W iff a f(a) f is a bijection, so W=f(a0), for some a0 A 2/22/12 a 0contradiction f(a0) iff a 0f(a ) 13 So ... every row! So cannot be listed: this diagonal sequence will be missing 2/22/12 10 Cantor’s Theorem For every set, A (finite or infinite), there is no bijection A↔P(A) 2/22/12 11 There is no bijection

Ngày tải lên: 22/03/2019, 10:49

... Example Induction Proof Let’s prove: 1+r +r + (for r ≠ 1) n +r = (n+ 1) r -1 r -1 Example Induction Proof Statements in magenta form a template for inductive proofs: • • Proof: (by induction ... +r n = (for r ≠ 1) ( n+ ) r -1 r -1 Example Induction Proof Base Case (n = 0): ? r 0+1 - 1+r +r +L +r = r -1 OK! r -1 = =1 r -1 Example Induction Proof • Inductive Step: Assume P(n) for some

Ngày tải lên: 22/03/2019, 10:51

... n-1 multiplications • Method 2: use successive squaring – How many times can you divide n by before it is reduced to 1? – Repeated squaring requires between log2n and 2∙log2n multiplications ... multiplications • Method 2: use successive squaring – Requires about log2n multiplications • Same idea works for multiplication modulo p • Example: If n is a 500-digit number, we can compute qn (mod p) in

Ngày tải lên: 22/03/2019, 10:55

...  ⎠ For example, n                  Note that n2+1 is being used to name the function f such that f(n) = n2+1 for every n 3/26/12 An example: Stirling’s formula ...   Little-Oh: f = o(g) • Def: f(n) = o(g(n)) if  lim  n   • For example, n2 = o(n3) since   lim      n     3/26/12 = o( ∙ ) is “all one symbol” ...                    So,? ?for? ?example, 3n      3/26/12  Rough Paraphrase • f∼g: f and g grow to be roughly equal •

Ngày tải lên: 22/03/2019, 10:59

... 4-colorable 1850’s: false proof published (was correct for colors) 1970’s: proof with computer 1990’s: much improved 3/16/12 15 Chromatic Number #colors for G is chromatic number, χ(G) lemma: 3/16/12 ... slots How short an exam period? 3/16/12 Harvard’s Solution Different “exam group” for every teaching hour Exams for different groups at different times 3/16/12 10 3/16/12 11 But This May be Suboptimal

Ngày tải lên: 22/03/2019, 11:07

... independent events iff Pr(A|B) = Pr(A) • Proof: That is, knowing whether B is the case gives no information that would help determine the probability of A A and B independent iff Pr(A)∙Pr(B) = Pr(A∩B) ... the probability that Santorum will be the Republican nominee? Total Probability Simple version: For any events A and B whose probability is neither nor 1: Pr(A) = Πρ( Α | Β)⋅ Πρ( Β) + Πρ( Α |

Ngày tải lên: 22/03/2019, 11:14

... one choice for the last So by the product rule, n ∙ (n-1) ∙ (n-2) ∙ … ∙ 2.1 = n! How Many 4-Letter Words Using Each Letter at Most Once? • • • • • • 26 choices for first letter Only 25 for second ... Letter at Most Once? • • • • • • 26 choices for first letter Only 25 for second letter 24 for third letter 23 for fourth letter So 26∙25∙24∙23 or 26!/22! Generalized Product Rule • • Let Q be a set ... |Q| = n1⋅n2⋅⋅⋅nk if n1 possible 1st elements, n2 possible 2nd elements (for each first entry), n3 possible 3rd elements (for each 1st & 2nd entry, ) then, How Many Hands with Cards? • • • • I.e.,

Ngày tải lên: 22/03/2019, 11:18

... transitive if R = G+ for some digraph G 3/22/19 Transitive Closure G+ is the transitive closure of the binary relation G 3/22/19 reflexivity relation R on set A is reflexive if a R a for all a A ≤ ... is reflexive if a R a for all a A ≤ on numbers and ⊆ on sets are reflexive 3/22/19 reflexivity For any digraph G, * G is reflexive 3/22/19 Reflexive Transitive Closure G* is the reflexive transitive ... IMPLIES NOT(v D 3/22/19 * u) 12 antisymmetry antisymmetric relation R: u R v IMPLIES NOT(v R u) for any u ≠ v If D is a DAG then * 3/22/19 13 (weak) partial orders Reflexive, Transitive, and Antisymmetric

Ngày tải lên: 22/03/2019, 11:30

... Directed Graphs 3/6/12 Normal Person’s Graph y = f(x) y x 3/6/12 Computer Scientist’s Graph a b c d f e 3/6/12 Digraphs • a set, V, of vertices aka nodes a set, ... v 3/6/12 w Relations and Graphs a b d c V= {a,b,c,d} E = {(a,b), (a,c), (c,b)} 3/6/12 Digraphs Formally, a digraph with vertices V is the same as a binary relation on V 3/6/12 Walks & Paths Walk:

Ngày tải lên: 22/03/2019, 11:32

... easy on a computer -about 100 digit-by-digit multiplications • What are the factors of 206085796112139733547? – Seems to require vast numbers April 25, 2012 10 Recall there’s a shortcut for computing ... it takes n steps to search through the first n possible exponents • For 500-digit numbers, we’re talking about a computing effort of 1700 steps vs 10500 steps April 25, 2012 13 Discrete logarithm ... these, and get at their meaning, he must substitute the fourth letter of the alphabet, namely D, for A, and so with the others.” April 25, 2012 Tyepmg Pic Gvctxskvetlc April 25, 2012 Public Key

Ngày tải lên: 22/03/2019, 11:50

... X(s) = if the flip comes up heads, if it comes up tails Example: S = Harvard basketball games, and for any game s∈S, X(s) = if Harvard wins game s, if Harvard loses These are examples of Bernoulli ... games, X(s) = number of points player LR scored in game s Probability Mass Function • • • • • For any x∈T, Pr({s∈S: X(s) = x}) is a well defined probability (Min 0, max 1, sum to over all possible

Ngày tải lên: 22/03/2019, 11:52

... from numbers to numbers in terms of the value of the same function for smaller arguments • Eg Fibonacci: – F0 = 0, F1 = 1, and for n>1, – Fn = Fn-1+Fn-2 A note on Fibonacci • Incredibly, the Fibonacci ... conquer • Determine whether an item x is in a sorted list L by binary search • For convenience assume list L has n elements for some n • Algorithm: – If L is of length 1, check to see if the unique ... is of length 2n+1 where n ≥ 0, compare x to L[2n] – If x≤L[2n] then search for x in L[1 2n] – If x>L[2n] then search for x in L[2n+1 2n+1] Analysis • Let Dn = # of comparison steps to find an

Ngày tải lên: 22/03/2019, 11:58

... Stacking books Can you stack identical books so the top one is completely off the table? 3/22/19 Stacking books • Can you stack identical books so the top one is completely off the table? • One book: ... middle 3/22/19 Stacking books • Two books: balance top one at the middle, the second over the table edge at the center of gravity of the pair 1 ½ 3/22/19 Stacking books  Three books: balance top ...     So with a stack of 31 books you get book lengths off the table! And there is no limit to how far the stack can extend! http://mathforum.org/advanced/robertd/harmonic.html 3/22/19

Ngày tải lên: 22/03/2019, 12:02

... ∑ n i=1 • For example, X = 1, 3, 2, 4, 1, 4, , then μX=16/7 • Note that the mean need not be one of the data values • We might as well write this as E[X] following the notation used for random ... same way, even though neither is exactly a function of the other • For example,Weight height and weight of people Height Covariance for Random Variables • Roll two dice Let X = larger of the two ... Probability and Statistics Probability vs Statistics • In probability, we build up from the mathematics of permutations and combinations and set theory a mathematical theory of how outcomes

Ngày tải lên: 22/03/2019, 12:06