 9.5.1E: a. List all 2combinations for the set {x1, x2, x3}. Deduce the val...
 9.5.2E: a. List all 3combinations for the set {x1, x2, x3, x4, x5}. Deduce...
 9.5.3E: Write an equation relating P(7, 2) and .
 9.5.4E: Write an equation relating P(8, 3) and .
 9.5.5E: Use Theorem to compute each of the following.a. ________________b. ...
 9.5.6E: A student council consists of 15 students.a. In how many ways can a...
 9.5.7E: A computer programming team has 13 members.a. How many ways can a g...
 9.5.8E: An instructor gives an exam with fourteen questions. Students are a...
 9.5.9E: A club is considering changing its bylaws. In an initial straw vote...
 9.5.10E: Two new drugs are to be tested using a group of 60 laboratory mice,...
 9.5.11E: Refer to Example 1. For each poker holding below, (1) find the numb...
 9.5.12E: How many pairs of two distinct integers chosen from the set {1, 2, ...
 9.5.13E: A coin is tossed ten times. In each case the outcome H (for heads) ...
 9.5.14E: a. How many 16bit strings contain exactly seven 1’s?b. How many 16...
 9.5.15E: a. How many even integers are in the set {1, 2, 3,… 100}?b. How man...
 9.5.16E: Suppose that three computer boards in a production run of forty are...
 9.5.17E: Ten points labeled A, B, C, D, E, F, G, H, I, J are arranged in a p...
 9.5.18E: Suppose that you placed the letters in Example 1 into positions in ...
 9.5.19E: a. How many distinguishable ways can the letters of the word HULLAB...
 9.5.20E: a. How many distinguishable ways can the letters of the word MILLIM...
 9.5.21E: In Morse code, symbols are represented by variablelength sequences...
 9.5.22E: Each symbol in the Braille code is represented by a rectangular arr...
 9.5.23E: On an 8 × 8 chessboard, a rook is allowed to move any number of squ...
 9.5.24E: The number 42 has the prime factorization 2. 3. 7. Thus 42 can be w...
 9.5.25E: a. How many onetoone functions are there from a set with three el...
 9.5.26E: a. How many onto functions are there from a set with three elements...
 9.5.27E: Let A be a set with eight elements.a. How many relations are there ...
 9.5.28E: A student council consists of three freshmen, four sophomores, four...
 9.5.29E: An alternative way to derive Theorem uses the following division ru...
 9.5.30E: Find the error in the following reasoning: “Consider forming a poke...
 9.5.31E: Let Pn be the number of partitions of a set with n elements. Show t...
 9.5.32E: Exercises 32–38 refer to the sequence of Stirling numbers of the se...
 9.5.33E: Exercises 32–38 refer to the sequence of Stirling numbers of the se...
 9.5.34E: Exercises 32–38 refer to the sequence of Stirling numbers of the se...
 9.5.35E: Exercises 32–38 refer to the sequence of Stirling numbers of the se...
 9.5.36E: Exercises 32–38 refer to the sequence of Stirling numbers of the se...
Solutions for Chapter 9.5: Discrete Mathematics with Applications 4th Edition
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 9.5
Get Full SolutionsThis 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. Since 36 problems in chapter 9.5 have been answered, more than 56524 students have viewed full stepbystep solutions from this chapter. Chapter 9.5 includes 36 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!

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

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

Dimension of vector space
dim(V) = number of vectors in any basis for V.

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

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

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.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

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

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

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

Solvable system Ax = b.
The right side b is in the column space of A.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

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

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

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