discrete mathematics for computer science solutions manual

... Trang 1Framingham State University 2/14 Study questions on Discrete Structures for Computer Science The intention in providing these questions is to show the student what ... table contains (a) variables; (b) formulas; (c) values of formulas under one interpretation; (d) values of formulas under all interpretations; (e) operations 17 If formulas  and  have the same ... (d) variables; (e) quantifiers 10 A formula in logic is valid if (a) it is true for some interpretation; (b) it is true for all interpretations; (c) it is true for no interpretation; (d) it is an

Ngày tải lên: 17/11/2016, 22:23

... steam-curing at 60oC for 24 hours In order to optimize the usage of formwork, the products were cast and steam-cured initially for about 4 hours The steam-curing was then stopped for some time to ... the case of reinforced Portland cement concrete columns and beams are applicable for reinforced geopolymer concrete columns Mid-span deflection at service load of Trang 8reinforced geopolymer ... manufactured ready for use within 24 hours after casting DESIGN OF GEOPOLYMER CONCRETE MIXTURES Concrete mixture design process is vast and generally based on performance criteria Based on the information

Ngày tải lên: 09/12/2016, 07:56

... semester • Function is extremely important in mathematics and computer science • linear, polynomial, exponential, logarithmic, • Don’t worry! For discrete mathematics, we need to understand functions ... Onto Functions Sequences and Summation Recursion 4.1 Chapter 4 Sets and Functions Discrete Structures for Computer Science (CO1007) on Ngày Trang 2Nguyen An Khuong, Huynh Tuong NguyenContents Sets ... which often have thesame properties Example • Set of all the students who are currently taking Discrete Mathematics 1 course • Set of all the subjects that K2011 students have to take in the first

Ngày tải lên: 29/03/2017, 18:30

... Partial Order Relations • Order words such that x comes before y in the dictionary • Schedule projects such that x must be completed before y • Order set of integers, where x < y Definition ... If u is an element of S such that a 4 u for all elements a ∈ A, then u is called anupper bound (cận trên) of A • If l is an element of S such that l 4 a for all elements a ∈ A, then l is called ... Equivalence Relations and Partitions Theorem Let R be an equivalence relation on a set A These statements for elements a and b of A are equivalent: i aRb ii [a] = [b] iii [a] ∩ [b] 6= ∅ Trang 31Nguyen An

Ngày tải lên: 29/03/2017, 18:30

... project from one of three fields: Information system (32 projects), Software Engineering (12 projects) and Computer Science (15 projects) How many ways are there for a student to choose? Solution: ... playing card, gambling, dices, • How many allowable passwords on a computer system? • How many ways to choose a starting line-up for a football Trang 4Nguyen An Khuong, Huynh Tuong NguyenContents ... & Combinations Example Question: How many ways we can choose3 students fromthe facultiesof Computer Science, Electrical Engineering and Mechanical Engineering? Trang 24Nguyen An Khuong, Huynh

Ngày tải lên: 29/03/2017, 18:30

... ⇒ ν2 = 4 κ2, ωηιχη ισ εϖεν Trang 4More 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 ... Q ” or “ P⇔Q ” or “ P ≡Q ” Trang 5More slowly …• Thm For any integer n, n2 is odd if and only if n is odd. (<=) If n is odd then n=2k+1 for some integer k … then n2=4k2+4k+1, which is odd ... (“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 Trang 6Contrapositive and converse• The contrapositive of

Ngày tải lên: 22/03/2019, 10:35

... bottom 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 ... 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} (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 Trang 10Why does this take 2n-3 flips? • m≠2 since one flip sorts 2 pancakes

Ngày tải lên: 22/03/2019, 10:37

... 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 determine whether a formula is a tautology ... Trang 1Logic and computersTrang 2Binary Arithmetic0 +0 0 0 +1 1 1 +0 1 1 +1 10 Only two digits: the bits 0 and 1 (Think: 0 = F, 1 = T) Trang 3Logic and Computers A half adder: ... number of bits to be added? 0 a2 b2 a1 b1 c2 c1 carryout Trang 12Simplifying Circuits• Simpler formulas turn into circuits that use less hardware! • E.g p ⋁ q ⋁ (p⋀q) is equivalent to p ⋁ q

Ngày tải lên: 22/03/2019, 10:41

... 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” ... Trang 1SetsTrang 2What 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

... Total f: R ×R → R f(x,y) = x/y Defined for all pairs (x,y) except when y=0! f Trang 6A 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! ... 10Cardinality or “Size”f For finite sets, a bijection exists iff A and B have the same number of elements Trang 11Cardinality or “Size”Use the same as a definition of “same size” for infinite sets:

Ngày tải lên: 22/03/2019, 10:47

... 1 1 Trang 11Cantor’s TheoremFor every set, A (finite or infinite) , there is no bijection 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) ... , for some a0 A (∀a) a f(a0) iff a  f(a ) Pf by contradiction: suppose 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 ... 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 f(a ) iff a  f(a ) Pf by contradiction: suppose f:A↔P(A) is a bijection Let

Ngày tải lên: 22/03/2019, 10:49

... level Fifth level Trang 5Example Induction ProofLet’s prove: r-1 (for r ≠ 1) Trang 6Statements in magenta form atemplate for inductive proofs: • Proof: (by induction on n ) • The induction ... 1+r +r + +r = r-1 L (for r ≠ 1) Trang 7Base Case (n = 0):r - 1 = = r - 1 1 L 0+1 1+r +r + +r = r - 1 Example Induction Proof 1 OK! Trang 8• Inductive Step: Assume P ( n ) for some n ≥ 0 and prove

Ngày tải lên: 22/03/2019, 10:51

... n-1 multiplications • Method 2: use successive squaring – How many times can you divide n by 2 before it is reduced to 1? – Repeated squaring requires between – Huge savings! n = 1000 => at ... 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 q n (mod p) in

Ngày tải lên: 22/03/2019, 10:55

... Functions3/26/12 Trang 2Asymptotic Equivalence• Def: For example, Note that n2 +1 is being used to name the function f such that f(n) = n2 +1 for every n f (n) : g(n) iff limn        ... example: Stirling’s formula n! : n e ⎛ ⎝⎜ ⎞ ⎠⎟    3/26/12 Trang 4Little-Oh: f = o(g)• Def: f(n) = o(g(n)) if lim n           • For example, n2 ...                So, for example, 3n2       Trang 11Rough Paraphrase• f∼g: f and g grow to be roughly equal •

Ngày tải lên: 22/03/2019, 10:59

... are independent events iff Pr(A|B) = Pr(A) • That is, knowing whether B is the case gives no information that would help determine the probability of A • Proof: A and B independent iff Pr(A)∙Pr(B) ... probability that Santorum will be the Republican nominee? Trang 9Total ProbabilitySimple version: For any events A and B whose probability is neither 0 nor 1: That is, Pr(A) is the weighted average

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! Trang 3How 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! Trang 4Generalized Product Rule• Let ... length-k sequences • if n1 possible 1st elements, n2 possible 2nd elements (for each first entry), n3 possible 3rd elements (for each 1st & 2nd entry, ) then, • |Q| = n1⋅n2⋅⋅⋅nk Trang 5How Many

Ngày tải lên: 22/03/2019, 11:18

... Trang 1Digraphs and RelationsTrang 5R is a transitive if R = G + for some digraph G transitivity Trang 8reflexivity For any digraph G, G* is reflexive Trang 9G* is the reflexive transitive ... is symmetric if a R b IMPLIES b R a Trang 14(weak) partial ordersTrang 15R is a W PO if R = D * for some DAG D weak partial orders Trang 16transitive, symmetric &reflexive equivalence relations

Ngày tải lên: 22/03/2019, 11:30

... Trang 1Directed GraphsTrang 2Normal Person’s Graphy y = f(x) Trang 3Computer Scientist’s Grapha f e c d b Trang 4• a set, V, of vertices aka “nodes” • a set, E ⊆ V×V ... notation: v→w Trang 5Relations and Graphsa c b d V= {a,b,c,d} E = {(a,b), (a,c), (c,b)} Trang 6Formally, a digraph with vertices V is the same as a binary relation on V Digraphs Trang 7Walks

Ngày tải lên: 22/03/2019, 11:32

... of the form qn = y (mod p) • Then it seems to be exponentially harder to compute n given y, than it is to compute y through the first n possible exponents. Discrete Logarithms Trang 14Discrete ... 28487532223✕72342452989? – Not hard easy on a computer about 100 digit-by-digit multiplications • What are the factors of 206085796112139733547? Trang 11Recall there’s a shortcut for computing powers • Problem: ... these, and get at their meaning, he must substitute the fourth letter of the alphabet, namely D, for Trang 3Tyepmg Pic GvctxskvetlcTrang 4Public Key CryptographyHow to Exchange Secrets in Public!

Ngày tải lên: 22/03/2019, 11:50

... original formula for q. Recall that our proof of the formula we had in Exercise 1.4-5 did not explain why the product of three factorials appeared in the denominator, it simply proved the formula ... distinct elements. There are n choices for the first number in the list. For each way of choosing the first element, there are n −1choices for the second. For each choice of the first two elements, ... the product in the denominator of the formula in Exercise 1.4-5 for the number of labellings with three labels is what it is, and could generalize this formula to four or more labels. Equivalence...

Ngày tải lên: 21/02/2014, 09:20