concrete mathematics a foundation for computer science epub

... special symbols, just as wehave adopted special notations for rising powers as well as falling powers.Mathematicians have long had both sine and cosine, tangent and cotangent,secant and cosecant, ... operations Mathematicians have used mod this way informally for a long time, taking various quantities mod 10, mod 277, and so on, but only in the last twenty years has it caught on formally ... something flaky about a sum that gives different values whenits terms are added up in different ways Advanced texts on analysis have a variety of definitions by which meaningful values can be assigned

Ngày tải lên: 14/08/2014, 04:21

... 12 and k = 8, we want to count thenumber of solutions to This means a0 = og = a4; al = a9 = as; a2 = alo = o6; and a3 = all = a7 So the values of ao, al, a2, and as can be chosen in n4 ways, and ... How can we capture this behavior in mathematical formulas that are easy to deal with? A bit of experimentation suggests that the best way is to maintain a 2 x 2 matrix that holds the four quantities ... symmetric law headache.”- The Hacker’s ‘a 3 b =$ b E a’, and the transitive law ‘a E b E c j a E c’ All these properties are easy to prove, because any relation ‘E’ that satisfies ‘a E b c J f(a) =

Ngày tải lên: 14/08/2014, 04:21

... theair The hats come back randomly, one hat to each of the n fans How manyways h(n, k) are there for exactly k fans to get their own hats back? For example, if n = 4 and if the hats and fans are ... - k)i, and we have the general formula h(n,k) = (Subfactorial notation isn’t standard, and it’s not clearly a great idea; but let’s try it awhile to see if we grow to like it We can always resort ... series was launched many years ago by ler, Gauss, and Riemann; such series, in fact, are still the subject of consid-erable research But hypergeometrics have a somewhat formidable notation, Eu-Anything

Ngày tải lên: 14/08/2014, 04:21

... overhang when n cards are stacked as described Could we achieve greater overhang by holding back, not pushing each card to an extreme position but storing up “potential gravitational energy” for a ... that H, is never an integer when n > 1.Here’s a card trick, based on an idea by R T Sharp [264], that illustrateshow the harmonic numbers arise naturally in simple situations Given n cardsand ... four cards And with 52 cards we have an H52-unit overhang, which turns out to be H52/2 x 2.27 cardlengths (We will soon learn a formula that tells us how to compute an approximate value of H, for

Ngày tải lên: 14/08/2014, 04:21

... the pattern.) There is no closed form for p(n), but the theory of partitions is a fascinating branch of mathematics in which many remarkable discoveries have been made For example, Ramanujan proved ... that formal power seriescan be manipulated algebraically without worrying about convergence di-Example 6: A recurrence that goes ail the way back. Let’s close this section by applying generating ... needs to be taken into account if we try to use formula (7.38) in numerical calculations For example, a fairly expensive name-brand hand Icalculator comes up with 413403.0005 when asked to compute

Ngày tải lên: 14/08/2014, 04:21

... means “mean.” “mean,” and “average” almost interchangeably in the rest of this chapter If X and Y are any two random variables defined on the same probability space, then X + Y is also a random ... l&k) [ACk’ =A[k)] (8.75) k=l Trang 24In the special case p = i we can interpret these formulas in a particularlysimple way Given a pattern A of m heads and tails, let A:A = fIkpl [Ack’ =A(kj] ... pretend that we don’t know anysuch formula; we merely want to study the average value of F,, and its stan-dard deviation The average value is, in fact, extremely easy to calculate, avoiding all thecomplexities

Ngày tải lên: 14/08/2014, 04:21

... logarithm that turns out to be very uncommon in mathematicsand computer science? Yes; and many mathematicians confuse the issue by using ‘log’ to stand for natural logarithms or binary logarithms ... ‘In in purely mathematical calculations, since the formulas fornatural logarithms are nice and simple But what about ‘log’? Isn’t thisthe “common” base-10 logarithm that students learn in high school-the ... paper again, gang. Trang 10444 ASYMPTOTICSProblem 4: A sum from an old final exam. When Concrete Mathematics was first taught at Stanford University dur-ing the 1970-1971 term, students were asked

Ngày tải lên: 14/08/2014, 04:21

... is equivalent to aa2-a’a+aa-a2+a+a > 0 , which is a consequence of aa( a - a) + (1 + a)a 3 ( 1 + a)a > a2 - a Hence we can replace x, and a by a - 1 and (3, repeating this transformation ... each pair aa becomes an multiplying Trang 29A ANSWERS TO EXERCISES 527If the harmonic aa and a bb Thus each bb behaves like a drone in the bee tree and each aa behaves like a queen, except that ... z) for 0 6 j 6 d, therebyeliminating an upper parameter and a lower parameter Thus, for example,we get closed forms for F( a, b; a - 1; z), F( a, b; a - 2; z), etc Gauss [116, $71 derived analogous

Ngày tải lên: 14/08/2014, 04:21

... 602. for 1x1,” American Mathematical Monthly 96 (1989), 525-526 131 Ronald L Graham, Donald E Knuth, and Oren Patashnik, Concrete 102.Mathematics: A Foundation for Computer Science Addison-Wesley, ... L Greitzer, International Mathematical Olympiads, 1959-1977 602. Mathematical Association of America, 1978 136 Oliver A Gross, “Preferential arrangements,” American Mathematical 604. Monthly 69 ... Nova acta academia: scientiarum Petropolitana: 9 (1791), 243-251 110 Martin Gardner, “About phi, an irrational number that has some re- 285. markable geometrical expressions,” Scientific American

Ngày tải lên: 14/08/2014, 04:21

... general says something quite different! Trang 7Proof by contradiction• To prove P, assume (not P) and show that a false statement logically follows • Then the assumption (not P) must have been ... Trang 82 is irrational• Suppose there were and derive a contradiction. m n    2 = 2 • That is, there are no integers m and n such that Trang 9• Suppose• Without loss of generality assume ... assume m and n have no common factors. – Because if both m and n were divisible by p, we could instead use and eventually find a fraction in lowest terms whose square is 2. 2 is irrational m n

Ngày tải lên: 22/03/2019, 10:35

... Given a stack of pancakes, make a nice stack with the smallest on top, then the next smallest, …, and the biggest on the bottom • By using only one operation: Grabbing a wad off the top and flipping ... pancake Trang 9Why 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 ... nonnegative integers has a least element arguing that a fraction can be reduced to “lowest terms” integer is nonempty Trang 6To prove P(n) for every nonnegative n: • Let C = {n: P(n) is false}

Ngày tải lên: 22/03/2019, 10:37

... (EQUIV) Trang 5Logic and Computers• S := A⊕B • C := A∧B A S B C Trang 6Half AdderA S B C HA A S B C Trang 7A Longer Addition11 +11 1 0 1 1 1 Trang 8Full Adder• Need a third input to create a component ... 10Full Adder Cin S A B Cout FA A B Cin S Cout HA HA Trang 11Ripple carry adder• 2-bit adder: a1a2+b1b2 = c1c2 with carryout • Generalizes to n-bit addition • How does the time delay through ... gates • But the P=NP? question means that it may be hard to simplify formulas as much as possible – Any tautology is equivalent to p ⋁ ¬p so if we could easily simplify formulas we could easily

Ngày tải lên: 22/03/2019, 10:41

... Trang 1SetsTrang 2What is a Set?• Informally, a collection of objects, determined by its members, treated as a single mathematical object • Not a real definition: What’s a collection?? Trang ... 7 ∈A • 8 ∉ A • N ∈ P(Z) Trang 6Subset: ⊆• A ⊆ B is read “A is a subset of B” 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) ... happens to be an infinite set) Trang 13Operators on Sets• Union: x∈A∪B iff x∈A or x∈B • Intersection: x∈A∩B iff x∈A and x∈B • Complement: x∈B iff x ∉ B • x∈A-B iff x∈A and x∉B • A-B = A\B = A∩B

Ngày tải lên: 22/03/2019, 10:45

... element of A is associated with at most one element of B a ⟼ b f(a) = b f AT MOST ONE ARROW OUT OF EACH ELEMENT OF A Trang 4Total Function: A → BEach element of A is associated with ONE AND ONLY ... a ⟼ b f(a) = b f EXACTLY ONE ARROW OUT OF EACH ELEMENT OF A Trang 5A Function that is “Partial,” Not Total f: R ×R → R f(x,y) = x/y Defined for all pairs (x,y) except when y=0! f Trang 6A ... element of A and exactly one arrow in to each element of B” Trang 10Cardinality or “Size”f For finite sets, a bijection exists iff A and B have the same number of elements Trang 11Cardinality or

Ngày tải lên: 22/03/2019, 10:47

... A↔P(A) Trang 12There is no bijection A↔P(A)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 ) Pf by contradiction: ... f:A↔P(A) is a bijection Let Pf by contradiction: Trang 13There is no bijection A↔P(A)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 ... …} Trang 7 Uncountably Infinite SetsClaim: ::= { ∞ -bit strings} is uncountable What about infinitely long bit strings? Like infinite decimal fractions but with bits Trang 8Diagonal ArgumentsSuppose

Ngày tải lên: 22/03/2019, 10:49

... Trang 4Like Dominos…Click to edit Master text styles Second level Third level Fourth level Fifth level Trang 5Example Induction ProofLet’s prove: r-1 (for r ≠ 1) Trang 6Statements in magenta ... magenta form atemplate for inductive proofs: • Proof: (by induction on n ) • The induction hypothesis, P( n ) , is: Example Induction Proof 1+r +r + +r = r-1 L (for r ≠ 1) Trang 7Base Case (n ... Trang 1InductionTrang 2The Idea of Induction0, 1, 2, 3, 4, 5, … then you know that all the ints are red ! Trang 3Induction Rule R ( 0 ), R ( 1 ), R ( 2 ),…, R ( n ),… R ( 0 ) and ("

Ngày tải lên: 22/03/2019, 10:51

... months a new pair is born • Once fertile a pair produces a new pair every month • Rabbits always come in breeding pairs, and never die http://morrischia.com/david/portfolio/boozy/research/fibonacci's_20rabbits.html ... ( 1 ) & R ( 2 ) I MPLI ES R ( 3 ),K R ( 0 ) and (" n) ( R ( 0 ) &º & R ( n )fi R ( n+1 )) (" m) R ( m ) Trang 4Fibonacci Numbers• Start with a pair of rabbits • After 2 ... 11 and be bad) But the previous n-1 digits could be any good string of length n-1 There are Cn-1 strings like that Total = Cn+1 = Cn+Cn-1 0 1 x Trang 14Proof by Induction that Cn=Fn+2(Base cases)

Ngày tải lên: 22/03/2019, 10:53

... never have been found by a brute-force computer search! The symbols ∀ ( for all”) and ∃ (“there exists”) are called quantifiers. A quantifier is always followed by a variable (and perhaps an indication ... indication of what values that variable can take on) and then a predicate that typically involves that variable. The predicate may itself involve more quantifiers. Here are a couple examples of statements ... prove that some statement holds for all natural values of a variable. For example, here is a classic formula: 46 Number Theory I generated lots of amazing ideas. But this lecture is about one...

Ngày tải lên: 05/03/2014, 23:20

... universal idea. Taking a walk is a literal example, but so is cooking from a recipe, executing a computer program, evaluating a formula, and recovering from substance abuse. Abstractly, taking a step ... for at least one x 2 R. All these sentences “quantify” how often the predicate is true. Specifically, an assertion that a predicate is always true is called a universal quantification, and an assertion ... get around the ambiguity of English, mathematicians have devised a spe- cial language for talking about logical relationships. This language mostly uses ordinary English words and phrases such as...

Ngày tải lên: 23/03/2014, 23:20

... that in a science of mind” sense CS has always existed. the criteria current in any culture for science may change greatly, but there always has been and always will be a science which deals ... supplying a formalization of language has led to the formation of dozens of mutually antagonistic camps, whose basic conceptions are often couched in highly baroque mathematical formalisms? Can it ... little too far. His eventual aim was to explain logic in action using the same set of concepts as for biological adaptation and psychological processes. Above all, he insisted that Kantian categories...

Ngày tải lên: 03/04/2014, 12:07