 12.3.12.3.1: Let A = to, 1 1 } and B = {00, 01}. Find each of these sets. a) AB
 12.3.12.3.2: Show that if A is a set of strings, then A0 = 0A = 0.
 12.3.12.3.3: Find all pairs of sets of strings A and B for which AB = { l 0, Ill...
 12.3.12.3.4: Show that these equalities hold. a) {A}* = {A} b) (A*)* = A* for ev...
 12.3.12.3.5: Describe the elements of the set A * for these values of A. a) { l ...
 12.3.12.3.6: Let V be an alphabet, and let A and B be subsets of V*. Show that I...
 12.3.12.3.7: Let V be an alphabet, and let A and B be subsets of V* with A B. Sh...
 12.3.12.3.8: Suppose that A is a subset of V*, where V is an alphabet. Prove or ...
 12.3.12.3.9: Determine whether the string 111 01 is in each of these sets. a) {O...
 12.3.12.3.10: Determine whether the string 01001 is in each of these sets. a) to,...
 12.3.12.3.11: Determine whether each of these strings is recognized by the determ...
 12.3.12.3.12: Determine whether each of these strings is recognized by the determ...
 12.3.12.3.13: Determine whether all the strings in each of these sets are recogni...
 12.3.12.3.14: Show that if M = (S, /, f, so, F) is a deterministic finitestate au...
 12.3.12.3.15: Given a deterministic finitestate automaton M = (S, /, f, so, F), ...
 12.3.12.3.16: In Exercises 1622 find the language recognized by the given determ...
 12.3.12.3.17: In Exercises 1622 find the language recognized by the given determ...
 12.3.12.3.18: In Exercises 1622 find the language recognized by the given determ...
 12.3.12.3.19: In Exercises 1622 find the language recognized by the given determ...
 12.3.12.3.20: In Exercises 1622 find the language recognized by the given determ...
 12.3.12.3.21: In Exercises 1622 find the language recognized by the given determ...
 12.3.12.3.22: In Exercises 1622 find the language recognized by the given determ...
 12.3.12.3.23: Construct a deterministic finitestate automaton that recognizes th...
 12.3.12.3.24: Construct a deterministic finitestate automaton that recognizes th...
 12.3.12.3.25: Construct a deterministic finitestate automaton that recognizes th...
 12.3.12.3.26: Construct a deterministic finitestate automaton that recognizes th...
 12.3.12.3.27: Construct a deterministic finitestate automaton that recognizes th...
 12.3.12.3.28: Construct a deterministic finitestate automaton that recognizes th...
 12.3.12.3.29: Construct a deterministic finitestate automaton that recognizes th...
 12.3.12.3.30: Construct a deterministic finitestate automaton that recognizes th...
 12.3.12.3.31: Construct a deterministic finitestate automaton that recognizes th...
 12.3.12.3.32: Construct a deterministic finitestate automaton that recognizes th...
 12.3.12.3.33: Construct a deterministic finitestate automaton that recognizes th...
 12.3.12.3.34: Construct a deterministic finitestate automaton that recognizes th...
 12.3.12.3.35: Construct a finitestate automaton that recognizes the set of bit s...
 12.3.12.3.36: Construct a finitestate automaton with four states that recognizes...
 12.3.12.3.37: Show that there is no finitestate automaton with two states that r...
 12.3.12.3.38: Show that there is no finitestate automaton with three states that...
 12.3.12.3.39: Explain how you can change the deterministic finitestate automaton...
 12.3.12.3.40: Use Exercise 39 and finitestate automata constructed in Example 6 ...
 12.3.12.3.41: Use the procedure you described in Exercise 39 and the finitestate...
 12.3.12.3.42: Use the procedure you described in Exercise 39 and the finitestate...
 12.3.12.3.43: In Exercises 4349 find the language recognized by the given nondet...
 12.3.12.3.44: In Exercises 4349 find the language recognized by the given nondet...
 12.3.12.3.45: In Exercises 4349 find the language recognized by the given nondet...
 12.3.12.3.46: In Exercises 4349 find the language recognized by the given nondet...
 12.3.12.3.47: In Exercises 4349 find the language recognized by the given nondet...
 12.3.12.3.48: In Exercises 4349 find the language recognized by the given nondet...
 12.3.12.3.49: In Exercises 4349 find the language recognized by the given nondet...
 12.3.12.3.50: Find a deterministic finitestate automaton that recognizes the sam...
 12.3.12.3.51: Find a deterministic finitestate automaton that recognizes the sam...
 12.3.12.3.52: Find a deterministic finitestate automaton that recognizes the sam...
 12.3.12.3.53: Find a deterministic finitestate automaton that recognizes the sam...
 12.3.12.3.54: Find a deterministic finitestate automaton that recognizes the sam...
 12.3.12.3.55: Find a deterministic finitestate automaton that recognizes each of...
 12.3.12.3.56: Find a nondeterministic finitestate automaton that recognizes each...
 12.3.12.3.57: Show that there is no finitestate automaton that recognizes the se...
 12.3.12.3.58: a) Show that for every nonnegative integer k, Rk is an equivalence ...
 12.3.12.3.59: Show that there is a nonnegative integer n such that the set of ne...
 12.3.12.3.60: a) Show that s arid t are Oequivalent if and only if either both s...
 12.3.12.3.61: a) Show that if M is a finitestate automaton, then the quotient au...
 12.3.12.3.62: Answer these questions about the finitestate automaton M shown her...
Solutions for Chapter 12.3: Modeling Computation
Full solutions for Discrete Mathematics and Its Applications  6th Edition
ISBN: 9780073229720
Solutions for Chapter 12.3: Modeling Computation
Get Full SolutionsDiscrete Mathematics and Its Applications was written by Patricia and is associated to the ISBN: 9780073229720. Since 62 problems in chapter 12.3: Modeling Computation have been answered, more than 12005 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 12.3: Modeling Computation includes 62 full stepbystep solutions. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 6.

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

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.

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

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

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.

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.

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

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

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.

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

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.

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

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.
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