 13.2.1E: Draw the state diagrams for the finitestate machines with these st...
 13.2.3E: Find the output generated from the input string 01110 for the finit...
 13.2.5E: Find the output for each of these input strings when given as input...
 13.2.4E: Find the output generated from the input string 10001 for the finit...
 13.2.2E: Give the state tables for the finitestate machines with these stat...
 13.2.6E: Find the output for each of these input strings when given as input...
 13.2.7E: Construct a finitestate machine that models an old fashioned soda...
 13.2.8E: Construct a finitestate machine that models a newspaper vending ma...
 13.2.9E: Construct a finitestate machine that delays an input string two bi...
 13.2.11E: Construct a finitestate machine for the logon procedure for a com...
 13.2.10E: Construct a finitestate machine that changes every other bit, star...
 13.2.13E: Construct a finitestate machine for a toll machine that opens a ga...
 13.2.12E: Construct a finitestate machine for a combination lock that contai...
 13.2.15E: Construct a finitestate machine for a restricted telephone switchi...
 13.2.14E: Construct a finitestate machine for entering a security code into ...
 13.2.16E: Construct a finitestate machine that gives an output of 1 if the n...
 13.2.20E: Construct the state diagram for the Moore machine with this state t...
 13.2.18E: Construct a finitestate machine that determines whether the input ...
 13.2.19E: Construct a finitestate machine that determines whether the word c...
 13.2.17E: Construct a finitestate machine that determines whether the input ...
 13.2.21E: Construct the state table for the Moore machine with the state diag...
 13.2.22E: Find the output string generated by the Moore machine in Exercise 2...
 13.2.23E: Find the output string generated by the Moore machine in Exercise 2...
 13.2.24E: Construct a Moore machine that gives an output of 1 whenever the nu...
 13.2.25E: Construct a Moore machine that determines whether an input string c...
Solutions for Chapter 13.2: Discrete Mathematics and Its Applications 7th Edition
Full solutions for Discrete Mathematics and Its Applications  7th Edition
ISBN: 9780073383095
Solutions for Chapter 13.2
Get Full SolutionsThis textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 7. Since 25 problems in chapter 13.2 have been answered, more than 195427 students have viewed full stepbystep solutions from this chapter. Chapter 13.2 includes 25 full stepbystep solutions. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095. This expansive textbook survival guide covers the following chapters and their solutions.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

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

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

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

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

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.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

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

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Outer product uv T
= column times row = rank one matrix.

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

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.

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

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

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

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

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.