 12.1.1E: In 1 and 2 let ? = {x, y} be an alphabet.a. Let L1 be the language ...
 12.1.2E: In 1 and 2 let ? = {x, y} be an alphabet.a. Let L3 be the language ...
 12.1.3E: a. If the expression ab +cd+· in postfix notation is converted to i...
 12.1.4E: In 46, describe L1, L2, , and ()* for the given languages L1 and L...
 12.1.5E: In 46, describe L1, L2, , and ()* for the given languages L1 and L...
 12.1.6E: In 46, describe L1, L2, , and ()* for the given languages L1 and L...
 12.1.7E: In 7–9, add parentheses to make the order of precedence clear in th...
 12.1.8E: In 7–9, add parentheses to make the order of precedence clear in th...
 12.1.9E: In 7–9, add parentheses to make the order of precedence clear in th...
 12.1.10E: In 10–12 use the convention about order of precedence to eliminate ...
 12.1.11E: In 10–12 use the convention about order of precedence to eliminate ...
 12.1.12E: In 10–12 use the convention about order of precedence to eliminate ...
 12.1.13E: In 13–15 use set notation to derive the language defined by the giv...
 12.1.14E: In 13–15 use set notation to derive the language defined by the giv...
 12.1.15E: In 13–15 use set notation to derive the language defined by the giv...
 12.1.16E: In 16–18 write five strings that belong to the language defined by ...
 12.1.17E: In 16–18 write five strings that belong to the language defined by ...
 12.1.18E: In 16–18 write five strings that belong to the language defined by ...
 12.1.19E: In 19–21 use words to describe the language defined by the given re...
 12.1.20E: In 19–21 use words to describe the language defined by the given re...
 12.1.21E: In 19–21 use words to describe the language defined by the given re...
 12.1.22E: In 22–24 indicate whether the given strings belong to the language ...
 12.1.23E: In 22–24 indicate whether the given strings belong to the language ...
 12.1.24E: In 22–24 indicate whether the given strings belong to the language ...
 12.1.25E: In 25–27 find a regular expression that defines the given language....
 12.1.26E: In 25–27 find a regular expression that defines the given language....
 12.1.27E: In 25–27 find a regular expression that defines the given language....
 12.1.28E: Let r, s, and t be regular expressions over ?= {a, b}. In 28–30 det...
 12.1.29E: Let r, s, and t be regular expressions over ?= {a, b}. In 28–30 det...
 12.1.30E: Let r, s, and t be regular expressions over ?= {a, b}. In 28–30 det...
 12.1.31E: In 31–39 write a regular expression to define the given set of stri...
 12.1.32E: In 31–39 write a regular expression to define the given set of stri...
 12.1.33E: In 31–39 write a regular expression to define the given set of stri...
 12.1.34E: In 31–39 write a regular expression to define the given set of stri...
 12.1.35E: In 31–39 write a regular expression to define the given set of stri...
 12.1.36E: In 31–39 write a regular expression to define the given set of stri...
 12.1.37E: In 31–39 write a regular expression to define the given set of stri...
 12.1.38E: In 31–39 write a regular expression to define the given set of stri...
 12.1.39E: In 31–39 write a regular expression to define the given set of stri...
 12.1.41E: Write a regular expression to define the set of strings of 0’s and ...
Solutions for Chapter 12.1: Discrete Mathematics with Applications 4th Edition
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 12.1
Get Full SolutionsThis 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. Since 40 problems in chapter 12.1 have been answered, more than 51253 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4. Chapter 12.1 includes 40 full stepbystep solutions.

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.)

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

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.

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.

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

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

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.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

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

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

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.

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

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

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

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·

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.