 5.9.1E: Consider the set of Boolean expressions defined in Example 5.9.1. G...
 5.9.2E: Let S be defined as in Example 5.9.2. Give derivations showing that...
 5.9.3E: Consider the MIUsystem discussed in Example 5.9.3. Give derivation...
 5.9.4E: The set of arithmetic expressions over the real numbers can be defi...
 5.9.5E: Define a set S recursively as follows:I. BASE: 1 ? SII. RECURSION: ...
 5.9.6E: Define a set S recursively as follows:I. BASE: a ? SII. RECURSION: ...
 5.9.7E: Define a set S recursively as follows:I. BASE: ? SII. RECURSION: If...
 5.9.8E: Define a set S recursively as follows:I. BASE: 1 ? S, 2 ? S, 3 ? S,...
 5.9.9E: Define a set S recursively as follows:I. BASE: 1 ? S, 3 ? S, 5 ? S,...
 5.9.10E: Define a set S recursively as follows:I. BASE: 0 ? S, 5 ? SII. RECU...
 5.9.11E: Define a set S recursively as follows:I. BASE: 0 ? SII. RECURSION: ...
 5.9.13E: Consider the set P of parenthesis structures defined in Example 5.9...
 5.9.15E: Give a recursive definition for the set of all strings of 0’s and 1...
 5.9.16E: Give a recursive definition for the set of all strings of 0’s and 1...
 5.9.17E: Give a recursive definition for the set of all strings of a’s and b...
 5.9.18E: Give a recursive definition for the set of all strings of a’s and b...
 5.9.19E: Use the definition of McCarthy’s 91 function in Example 5.9.6 to sh...
 5.9.20E: Prove thatMcCarthy’s 91 function equals 91 for all positive integer...
 5.9.21E: Use the definition of the Ackermann function in Example 5.9.7 to co...
 5.9.22E: Use the definition of the Ackermann function to show the following:...
 5.9.23E: Compute T (2), T (3), T (4), T (5), T (6), and T (7) for the “funct...
 5.9.24E: Student A tries to define a function F : Z+ ? Z by the rule for all...
 5.9.25E: Student C tries to define a function G : Z+ ? Z by the rule for all...
Solutions for Chapter 5.9: Discrete Mathematics with Applications 4th Edition
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 5.9
Get Full SolutionsSince 23 problems in chapter 5.9 have been answered, more than 24240 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Discrete Mathematics with Applications was written by Sieva Kozinsky and is associated to the ISBN: 9780495391326. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4th. Chapter 5.9 includes 23 full stepbystep solutions.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Column space C (A) =
space of all combinations of the columns of 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.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Iterative method.
A sequence of steps intended to approach the desired solution.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

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.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

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.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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).

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

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

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.