 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 36444 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4. 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 and is associated to the ISBN: 9780495391326.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.