 13.1.1E: Use the set of productions to show that each of these sentences is ...
 13.1.2E: Find five other valid sentences, besides those given in Exercise 1.
 13.1.3E: Show that the hare runs the sleepy tortoise is not a valid sentence.
 13.1.4E: Let G = (V, T, S, P) be the phrasestructure grammar with V = {0. 1...
 13.1.6E: Let V = (S, A, B, a, b} and T = {a, b}. Find the language generated...
 13.1.5E: Let G = (V, T. S, P) be the phrasestructure grammar with V = {0,1,...
 13.1.7E: Construct a derivation of 0313using the grammar given in Example 5.
 13.1.8E: Show that the grammar given in Example 5 generates the set {0n 1n ...
 13.1.9E: a) Construct a derivation of 0214 using the grammar G1 in Example 6...
 13.1.10E: a) Show that the grammar G1 given in Example 6 generates the set {0...
 13.1.11E: Construct a derivation of 021222 in the grammar given in Example 7.
 13.1.12E: Show that the grammar given in Example 7 generates the set {0n1n2n ...
 13.1.13E: Find a phrasestructure grammar for each of these languages.a) the ...
 13.1.16E: Construct phrasestructure grammars to generate each of these sets....
 13.1.15E: Find a phrasestructure grammar for each of these languages.a) the ...
 13.1.14E: Find a phrasestructure grammar for each of these languages.a) the ...
 13.1.17E: Construct phrasestructure grammars to generate each of these sets....
 13.1.20E: A palindrome is a string that reads the same backward as it does fo...
 13.1.21E: Let G1 and G2 be contextfree grammars, generating the languages L(...
 13.1.18E: Construct phrasestructure grammars to generate each of these sets....
 13.1.19E: Let V = {S, A, B, a, b} and T = {a, b}. Determine whether G = (V, T...
 13.1.23E: Construct derivation trees for the sentences in Exercise 1.
 13.1.22E: Find the strings constructed using the derivation trees shown here.
 13.1.25E: Use topdown parsing to determine whether each of the following str...
 13.1.24E: Let G be the grammar with V = {a, b. c, S}; T = {a. b. c}; starting...
 13.1.28E: a) Explain what the productions are in a grammar if the BackusNaur...
 13.1.30E: a) Construct a phrasestructure grammar for the set of all fraction...
 13.1.29E: a) Construct a phrasestructure grammar that generates all signed d...
 13.1.27E: Construct a derivation tree for ?109 using the grammar given in Exa...
 13.1.26E: Use bottomup parsing to determine whether the strings in Exercise ...
 13.1.32E: Give production rules in BackusNaur form for the name of a person ...
 13.1.34E: Describe the set of strings defined by each of these sets of produc...
 13.1.31E: Give production rules in BackusNaur form for an identifier if it c...
 13.1.35E: Give production rules in extended BackusNaur form that generate al...
 13.1.39E: For each of these strings, determine whether it is generated by the...
 13.1.36E: Give production rules in extended BackusNaur form that generate a ...
 13.1.37E: Give production rules in extended BackusNaur form for identifiers ...
 13.1.42E: Let G be a grammar and let R be the relation containing the ordered...
 13.1.41E: For each of these strings, determine whether it is generated by the...
 13.1.38E: Describe how productions for a grammar in extended BackusNaur form...
 13.1.40E: Use BackusNaur form to describe the syntax of expressions in infix...
Solutions for Chapter 13.1: Discrete Mathematics and Its Applications 7th Edition
Full solutions for Discrete Mathematics and Its Applications  7th Edition
ISBN: 9780073383095
Solutions for Chapter 13.1
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 41 problems in chapter 13.1 have been answered, more than 187199 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 7. Chapter 13.1 includes 41 full stepbystep solutions. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

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.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

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

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

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).

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

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

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.

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

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

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

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

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.