 13.3.5E: Describe the elements of the set A* for these values of A. a) {10} ...
 13.3.6E: Let V be an alphabet, and let A and B be subsets of V*. Show that ...
 13.3.9E: Determine whether the string 11101 is in each of these sets.a) {0,1...
 13.3.7E: Let V be an alphabet, and let A and B be subsets of V* with Show that
 13.3.8E: Suppose that A is a subset of V*, where V is an alphabet. Prove or ...
 13.3.10E: Determine whether the string 01001 is in each of these sets.a) {0. ...
 13.3.11E: Determine whether each of these strings is recognized by the determ...
 13.3.14E: Show that if M = (S,I,f,s0,F) is a deterministic finitestate autom...
 13.3.15E: Given a deterministic finitestate automaton M = (S,I,f,s0,F), use ...
 13.3.12E: Determine whether each of these strings is recognized by the determ...
 13.3.13E: Determine whether all the strings in each of these sets are recogni...
 13.3.16E:
 13.3.17E:
 13.3.18E:
 13.3.19E:
 13.3.20E:
 13.3.21E:
 13.3.22E:
 13.3.23E: Construct a deterministic finitestate automaton that recognizes th...
 13.3.24E: Construct a deterministic finitestate automaton that recognizes th...
 13.3.25E: Construct a deterministic finitestate automaton that recognizes th...
 13.3.26E: Construct a deterministic finitestate automaton that recognizes th...
 13.3.27E: Construct a deterministic finitestate automaton that recognizes th...
 13.3.28E: Construct a deterministic finitestate automaton that recognizes th...
 13.3.29E: Construct a deterministic finitestate automaton that recognizes th...
 13.3.31E: Construct a deterministic finitestate automaton that recognizes th...
 13.3.30E: Construct a deterministic finitestate automaton that recognizes th...
 13.3.33E: Construct a deterministic finitestate automaton that recognizes th...
 13.3.36E: Construct a finitestate automaton that recognizes the set of bit s...
 13.3.35E: Construct a deterministic finitestate automaton that recognizes th...
 13.3.34E: Construct a deterministic finitestate automaton that recognizes th...
 13.3.38E: Show that there is no finitestate automaton with three states that...
 13.3.39E: Explain how you can change the deterministic finitestate automaton...
 13.3.37E: Show that there is no finitestate automaton with two states that r...
 13.3.41E: Use the procedure you described in Exercise 39 and the finitestate...
 13.3.40E: Use Exercise 39 and finitestate automata constructed in Example 6 ...
 13.3.42E: Use the procedure you described in Exercise 39 and the finitestate...
 13.3.43E:
 13.3.44E:
 13.3.45E:
 13.3.46E:
 13.3.47E:
 13.3.32E: Construct a deterministic finitestate automaton that recognizes th...
 13.3.51E: Find a deterministic finitestate automaton that recognizes the sam...
 13.3.48E:
 13.3.50E: Find a deterministic finitestate automaton that recognizes the sam...
 13.3.49E:
 13.3.52E: Find a deterministic finitestate automaton that recognizes the sam...
 13.3.53E: Find a deterministic finitestate automaton that recognizes the sam...
 13.3.55E: Find a deterministic finitestate automaton that recognizes each of...
 13.3.54E: Find a deterministic finitestate automaton that recognizes the sam...
 13.3.57E: Show that there is no finitestate automaton that recognizes the se...
 13.3.56E: Find a nondeterministic finitestate automaton that recognizes each...
 13.3.58E: In Exercises 58 we introduce a technique for constructing a determi...
 13.3.60E: In Exercises 60 we introduce a technique for constructing a determi...
 13.3.61E: In Exercises 61 we introduce a technique for constructing a determi...
 13.3.59E: In Exercises 59 we introduce a technique for constructing a determi...
 13.3.62: In Exercises 62 we introduce a technique for constructing a determi...
 13.3.2E: Show that if A is a set of strings, then A?= ?A = ?.
 13.3.4E: Show that these equalities hold.a){ ? }* = { ? }b)(A*)* = A* for ev...
 13.3.3E: Find all pairs of sets of strings A and B for which AB = {10. 1ll, ...
 13.3.1E: Let A = {0, 11} and B = {00,01}. Find each of these sets.a) ABb) BA...
Solutions for Chapter 13.3: Discrete Mathematics and Its Applications 7th Edition
Full solutions for Discrete Mathematics and Its Applications  7th Edition
ISBN: 9780073383095
Solutions for Chapter 13.3
Get Full SolutionsChapter 13.3 includes 62 full stepbystep solutions. Since 62 problems in chapter 13.3 have been answered, more than 186477 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 7. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

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.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

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.

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

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

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.

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

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

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

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.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

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