discrete mathematics and its applications induction

... Mathematical? ?Induction Strong? ?Induction Well­Ordering Recursive Definitions Structural? ?Induction Recursive Algorithms Program Correctness (not yet included in overheads) Mathematical Induction ... Section Summary Mathematical? ?Induction Examples of Proof by Mathematical? ?Induction Mistaken Proofs by Mathematical? ?Induction Guidelines for Proofs by Mathematical? ?Induction Climbing an Infinite ... Applying (2) again, the third rung.? ?And? ?so on.  We  can apply (2) any number of times to reach any  particular rung, no matter how high up This example motivates proof by  mathematical? ?induction Principle of Mathematical Induction

Ngày tải lên: 14/10/2020, 14:55

... “It is raining.” then p ∧q   denotes “I am at home? ?and? ?it is  Disjunction The disjunction of propositions  p ? ?and? ?q   is denoted by  p  ∨q? ?and? ?has this truth table: p q  p ∨q T T T T F T F T T F F F Example:  If p  denotes “I am at home.”? ?and? ?q  denotes  ... The conjunction of propositions  p ? ?and? ? q  is denoted by p  ∧ q ? ?and? ?has this truth table: p q p ∧ q  T T T T F F F T F F F F Example:  If p  denotes “I am at home.”? ?and? ?q  denotes  “It is raining.” then p ∧q   denotes “I am at home? ?and? ?it is  ... assignment of truth values to? ?its? ?variables that make it  true. When no such assignments exist, the compound  proposition is unsatisfiable A compound proposition is unsatisfiable if? ?and? ?only if? ?its? ? negation is a tautology

Ngày tải lên: 14/10/2020, 14:52

... x  (C(x )∨  y (C(y ) ∧ F(x, y)))       where C(x) is “x has a computer,”? ?and? ?F(x,y) is “x? ?and? ?y are  friends,”? ?and? ?the domain for both x? ?and? ?y consists of all  students in your school.      Solution: Every student in your school has a computer or has a  ...  At each step, loop through the values for y.    If for some pair of x andy, P(x,y) is false, then  x  yP(x,y) is  false? ?and? ?both the outer? ?and? ?inner loop terminate      x  y P(x,y) is true if the outer loop ends after stepping  ... meaning of sentences in English as well as important  concepts in computer science? ?and? ?mathematics.       Example: “Every real number has an inverse” is             x  y(x + y = 0)        where the domains of x? ?and? ?y are the real numbers We can also think of nested propositional functions:

Ngày tải lên: 14/10/2020, 14:52

... First try forward reasoning.  Start with the axioms? ?and? ? known theorems? ?and? ?construct a sequence of steps that  end in the conclusion.  Start with p? ?and? ?prove q, or start  with ¬q? ?and? ?prove ¬p If this doesn’t work, try backward reasoning. When trying  ... Key fact: Each domino covers one black? ?and? ?one white  square.  Therefore the tiling covers 31 black squares? ?and? ?31 white  squares Our board has either 30 black squares? ?and? ?32 white  squares or 32 black squares? ?and? ?30 white squares.   ... number √2 √2 . If it is rational, we have two irrational  numbers x? ?and? ?y with xy  rational, namely x = √2      ? ?and? ?y =  √2. But if √2 √2   is irrational,                              then we can  let  x = √2 √2 ? ?and? ?y = √2 so that             xy   = (√2 √2  )√2  = 

Ngày tải lên: 14/10/2020, 14:53

... Sequences? ?and? ?Summations Sets Section 2.1 Section Summary Definition of sets Describing Sets Roster Method Set­Builder Notation Some Important Sets in? ?Mathematics Empty Set? ?and? ?Universal Set ... Empty Set? ?and? ?Universal Set Subsets? ?and? ?Set Equality Cardinality of Sets Tuples Introduction Sets are one of the basic building blocks for the types of  objects considered in? ?discrete? ?mathematics Important for counting ... Basic Structures: Sets, Functions, Sequences, Sums, and Matrices Chapter With Question/Answer Animations Copyright © McGraw-Hill Education All rights

Ngày tải lên: 14/10/2020, 14:54

... Representations of integers, including binary? ?and? ? hexadecimal representations, are part of number theory.  Number theory has long been studied because of the  beauty of? ?its? ?ideas,? ?its? ?accessibility,? ?and? ?its? ?wealth of open  ... Divisibility? ?and? ?Modular Arithmetic Integer Representations? ?and? ?Algorithms  Primes? ?and? ?Greatest Common Divisors Solving Congruences  ? ?Applications? ?of Congruences Cryptography Divisibility and Modular ... If a | b, then a | bc for all integers c; iii If a | b? ?and? ?b | c, then a | c    Proof: (i)  Suppose a | b? ?and? ?a | c, then it follows that there are  integers s? ?and? ?t with b = as? ?and? ?c = at. Hence,             b + c = as + at = a(s + t).    Hence,  a | (b + c)

Ngày tải lên: 14/10/2020, 14:55

... The Pigeonhole Principle Permutations? ?and? ?Combinations Binomial Coefficients? ?and? ?Identities Generalized Permutations? ?and? ?Combinations Generating Permutations? ?and? ?Combinations (not yet  included in overheads) ... specifies that a telephone number consists of 10 digits,  consisting of a three­digit area code, a three­digit office  code,? ?and? ?a four­digit station code.  There are some  restrictions on the digits Let X denote a digit from 0 through 9 ... task? ?and? ?n2 ways to do the second task. Then there are n1∙n2  ways to do the procedure    Example: How many bit strings of length seven are there?    Solution: Since each of the seven bits is either a 0 or a 1, 

Ngày tải lên: 14/10/2020, 14:56

... linear. For example, the expected value of the sum of  random variables is the sum of their expected values.     Theorem 3: If Xi, i = 1, 2, …,n with n a positive integer,  are random variables on S,? ?and? ?if a? ?and? ?b are real numbers,  ...      Example: In a lottery, a player wins a large prize when  they pick four digits that match, in correct order, four digits  selected by a random mechanical process. What is the  probability that a player wins the  prize?  ... ways to pick four digits.  Since there is only 1 way to pick the correct digits, the  probability of winning the large prize is 1/10,000 = 0.0001      A smaller prize is won if only three digits are matched. 

Ngày tải lên: 14/10/2020, 14:56

... to another data center Section Summary Euler Paths? ?and? ?Circuits Hamilton Paths? ?and? ?Circuits ? ?Applications? ?of Hamilton Circuits Euler Paths and Circuits The town of Kӧnigsberg, Prussia (now Kalingrad, Russia)  ... G1 contains exactly two vertices of odd degree (b? ?and? ?d).  Hence it has an Euler path, e.g.,  d, a, b, c, d, b   Applications of Euler Paths and Circuits Euler paths? ?and? ?circuits can be used to solve many  practical problems such as finding a path or circuit that  ... link in a communications network Other? ?applications? ?are found in the  layout of circuits,  William Rowan  Hamilton (1805­  1865) Hamilton Paths and Circuits Euler paths? ?and? ?circuits contained every edge only once. 

Ngày tải lên: 14/10/2020, 14:57

... Theorem: An undirected graph is a tree if? ?and? ?only if there  is a unique simple path between any two of? ?its? ?vertices.  Proof: Assume that T is a tree. Then T is connected with no  simple circuits. Hence, if  x? ?and? ?y are distinct vertices of T,  ... Proof: Suppose that a simple graph G has a spanning tree T.  T contains every vertex of G? ?and? ?there is a path in T  between any two of? ?its? ?vertices. Because T is a subgraph of  G, there is a path in G between any two of? ?its? ?vertices.  Hence, G is connected.  ... Form a path starting at this vertex by successively adding  vertices? ?and? ?edges, where each new edge is incident with  the last vertex in the path? ?and? ?a vertex not already in the  path. Continue adding vertices? ?and? ?edges to this path as long 

Ngày tải lên: 14/10/2020, 14:58

... Examples of Circuits Logic Gates We construct circuits using gates, which take as input the  values of two or more Boolean variables? ?and? ?produce one  or more bits as output,? ?and? ?inverters, which take the value  ... All  identities in Table 5, except for  the first? ?and? ?the last two come in  pairs. Each element of the pair is the  dual of the other (obtained by  switching Boolean sums? ?and? ? Boolean products? ?and? ?0’s? ?and? ?1’s The Boolean identities correspond to the  ... (continued) A full adder is used to compute the sum bit? ?and? ?the carry  bit when two bits? ?and? ?a carry are added Adders (continued) A half adder? ?and? ?multiple full adders can be used to  produce the sum of n bit integers. 

Ngày tải lên: 14/10/2020, 14:58

... We use an auxiliary tape symbol M as a marker,? ?and? ?specify  that  V = {0, 1}? ?and? ?   I = {0, 1, M}.  Our TM has one final state, s6 . The TM successively  replaces a 0 at the leftmost position of the string with an M  and? ?a 1 at the rightmost position of the string with an M,  ... The machine starts at the leftmost 1 of the input string,   and? ?proceeds to erase this 1.  If the next character is an asterisk, n1 = 0. In this case, it  replaces the asterisk with a blank? ?and? ?halts Otherwise, it erases the next 1,? ?and? ?then passes over the  ... concept of solvability by an effective algorithm is  informal? ?and? ?imprecise, as opposed to the concept of  solvability by a TM, which is formal? ?and? ?precise Decidability and Complexity A decision problem asks whether statements from a 

Ngày tải lên: 14/10/2020, 14:58

... q, ¬, and the connective ∨ that has this truth table p T T F F q T F T F ? F F T F 11 Find a proposition with three variables p, q, and r that is true when p and r are true and q is false, and ... combinatorial circuits in 54–55 54 p q r 55 p q q r Construct a combinatorial circuit using inverters, OR gates, and AND gates, that produces the outputs in 56–57 from input bits p, q and r 56 (¬p ... 10}, and f and g are defined by g = {(1, b), (2, a), (3, a), (4, b)} and f = {(a, 10), (b, 7), (c, 2)} Find f −1 486 Test Bank Questions and Answers In questions 134–137 suppose that g: A → B and

Ngày tải lên: 24/10/2017, 14:35

... positive), and ac > bc (when c is negative) General forms: cx + dy £ e or cx + dy ³ e where c and d are not both Standard forms: y £ mx + b, y ³ mx + b, x £ a, and x ³ a 10 Put the inequality into standard ... intercepts are (0, 4) and (1.6, 0) When x = then y = –1 b The intercepts are at the points (2, 0) and (0, 6) c When x = 2, y = 74 a y = 25x – 77 2y + x = 100 When y = 0, x = 100, and when x = 0, y ... [-10, 110] and [-10,60] Other answers are possible b c 75 a (0, –2) and ( 8,0) are intercepts When x = 2, y = –1.5 3y - 2x = 3y = 2x + y= x+3 78 x – 3y = 60 When x = 0, then y = - 20 and when y

Ngày tải lên: 20/08/2020, 11:45

... following published and/ or accepted journal and conference papers: 1 Anh T N and Hosoda T.: Depth-Averaged model of open channel flows over an arbitrary 3D surface and its applications to analysis ... channel, flow in meandering channel, and flow in Parshall flume with supercritical outflow was satisfactory Zhou and Goodwill (1997) presented a depth-averaged model and compared its predictions ... almost meandering and have the irregular boundaries, a “stair stepped” approximation is commonly used by the conventional finite-difference method and finite volume method (Vreugdenhil and Wijbenga

Ngày tải lên: 06/11/2012, 10:35

... MASS TRANSFER IN MULTIPHASE SYSTEMS AND ITS APPLICATIONS Edited by Mohamed El-Amin Mass Transfer in Multiphase Systems and its Applications Edited by Mohamed El-Amin Published by ... side) and between non-wetting and solid phase (right hand side) after 17 s 17 17 Mass and Heat Transfer During Two-Phase Flow in Porous Media - Theory and Modeling Mass and ... hand side) and heat transfer across fluid–fluid as well as fluid–solid interfaces (right hand side) 14 14 Mass Transfer. .. differences between wetting and non-wetting phase (left hand

Ngày tải lên: 20/06/2014, 06:20

... Transfer in Multiphase Systems and its Applications Solute Transport With Chemical Reaction in Single- and Multi-Phase Flow in Porous Media 9 Fig. 4. Physical model and coordinate system. Momentum: ... quantities of α y and D y are variables defined as α y = α + γd | v | and D y = D + ζd | v | where, α and D are the molecular thermal and solutal diffusivities, respectively, whereas γd | v | and ζd | ... approximations, and defining the velocity components u and v in terms of stream function ψ as: u = ∂ψ/∂y and v = −∂ψ/ ∂x, the pressure term may be eliminated between Eqs. (17) and (18) and one obtains:

Ngày tải lên: 20/06/2014, 06:20

... in Multiphase Systems and its Applications and density ρp = 2600 kg/m3 The measurements were conducted under various aeration rates, impeller speeds, and solid particles concentration ... watch its propagation throughout the cell volume, to estimate the intensity of its variation using the rate of change of the interference bands at the selected points (e.g. in drop center) and ... the drop or as periodic up -and- down motions of the emulsion over the drop surface (at С d ~1%) (Kostarev, 2005). Mass Transfer in Multiphase Systems and its Applications 74 a) d)

Ngày tải lên: 20/06/2014, 06:20

... scientific challenge and creates a 138 138 Mass Transfer in Multiphase Systems and its Applications Mass Transfer in Multiphase Systems and its Applications major ... 140 140 Mass Transfer in Multiphase Systems and its Applications Mass Transfer in Multiphase Systems and its Applications. .. aC b dt dt Where a is the interfacial ... Multiphase Systems and its Applications 110 The values of the coefficient A, m 1 , m 2 , and exponents B, C in this Eq. are collected in Table 6 for single impeller system and in Table 7 for

Ngày tải lên: 20/06/2014, 06:20

... Systems and its Applications 156 2. Analogy for mass transfer and wall shear stress in two-phase flow 2.1 Introduction The combined flow of gas and liquid intensifies significantly the heat and ... (2.Tw + Tin)/3 and ωref = (2.ωw + in) /3, where Tw , Tin, ωw and in are respectively the channel wall and 182 Mass Transfer in Multiphase Systems and its Applications inlet ... and its Applications Lockhart, R & Martinelli, R (1949) Proposed correlation of data for isothermal two-phase, two—component flow in pipes,... Multiphase Systems and its Applications

Ngày tải lên: 20/06/2014, 06:20