 9.2.1E: In, use the fact that in baseball’s World Series, the first team to...
 9.2.2E: In, use the fact that in baseball’s World Series, the first team to...
 9.2.3E: In, use the fact that in baseball’s World Series, the first team to...
 9.2.4E: In, use the fact that in baseball’s World Series, the first team to...
 9.2.5E: In a competition between players X and Y, the first player to win t...
 9.2.6E: One urn contains two black balls (labeled B1 and B2) and one white ...
 9.2.7E: One urn contains one blue ball (labeled B1) and three red balls (la...
 9.2.8E: A person buying a personal computer system is offered a choice of t...
 9.2.9E: Suppose there are three roads from city A to city B and five roads ...
 9.2.10E: Suppose there are three routes from North Point to Boulder Creek, t...
 9.2.11E: a. A bit string is a finite sequence of 0’s and 1’s. How many bit s...
 9.2.12E: Hexadecimal numbers are made using the sixteen digits 0, 1, 2, 3, 4...
 9.2.13E: A coin is tossed four times. Each time the result H for heads or T ...
 9.2.14E: Suppose that in a certain state, all automobile license plates have...
 9.2.15E: A combination lock requires three selections of numbers, each from ...
 9.2.16E: a. How many integers are there from 10 through 99?b. How many odd i...
 9.2.17E: a. How many integers are there from 1000 through 9999?b. How many o...
 9.2.18E: The diagram below shows the keypad for an automatic teller machine....
 9.2.19E: Three officers?a president, a treasurer, and a secretary?are to be ...
 9.2.20E: Modify Example 1 by supposing that a PIN must not begin with any of...
 9.2.21E: Suppose A is a set with m elements and B is a set with n elements.a...
 9.2.22E: a. How many functions are there from a set with three elements to a...
 9.2.23E: In Section 2.5 we showed how integers can be represented by strings...
 9.2.24E: In each of 24–28, determine how many times the innermost loop will ...
 9.2.25E: In each of 24–28, determine how many times the innermost loop will ...
 9.2.26E: In each of 24–28, determine how many times the innermost loop will ...
 9.2.27E: In each of 24–28, determine how many times the innermost loop will ...
 9.2.28E: In each of 24–28, determine how many times the innermost loop will ...
 9.2.29E: Consider the numbers 1 through 99,999 in their ordinary decimal rep...
 9.2.30E: Let where p1, p2,…, pm are distinct prime numbers and k1, k2, ..., ...
 9.2.31E: a. If p is a prime number and a is a positive integer, how many dis...
 9.2.32E: a. How many ways can the letters of the word ALGORITHM be arranged ...
 9.2.33E: Six people attend the theater together and sit in a row with exactl...
 9.2.34E: Five people are to be seated around a circular table. Two seatings ...
 9.2.35E: Write all the 2permutations of {W, X, Y, Z}.
 9.2.36E: Write all the 3permutations of {s, t, u, v}.
 9.2.37E: Evaluate the following quantities.a. P(6, 4)________________b. P(6,...
 9.2.38E: a. How many 3permutations are there of a set of five objects?b. Ho...
 9.2.39E: a. How many ways can three of the letters of the word ALGORITHM be ...
 9.2.40E: Prove that for all integers n ? 2, P(n + 1, 3) = n3 – n.
 9.2.41E: Prove that for all integers n ? 2,P(n + 1, 2) – P(n, 2) = 2P(n, 1).
 9.2.42E: Prove that for all integers n ? 3,P(n + 1, 3) – P(n, 3) = 3P(n, 2).
 9.2.43E: Prove that for all integers n ? 2, P(n, n) = P(n, n – 1).
 9.2.44E: Prove Theorem by mathematical induction.TheoremThe Multiplication R...
 9.2.45E: Prove Theorem by mathematical induction.TheoremFor any integer n wi...
 9.2.46E: Prove Theorem by mathematical induction.TheoremIf n and r are integ...
 9.2.47E: A permutation on a set can be regarded as a function from the set t...
Solutions for Chapter 9.2: Discrete Mathematics with Applications 4th Edition
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 9.2
Get Full SolutionsSince 47 problems in chapter 9.2 have been answered, more than 24930 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4th. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 9.2 includes 47 full stepbystep solutions. Discrete Mathematics with Applications was written by Sieva Kozinsky and is associated to the ISBN: 9780495391326.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).