 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 43336 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 and is associated to the ISBN: 9780495391326. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4. Chapter 5.9 includes 23 full stepbystep solutions.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

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!

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

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

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

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

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

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

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

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.

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

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

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

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

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).