 12.1.1: a. Let L1 be the language consisting of all strings over that are p...
 12.1.2: a. Let L3 be the language consisting of all strings over of length ...
 12.1.3: a. If the expression ab + cd + in postfix notation is converted to ...
 12.1.4: L1 is the set of all strings of as and bs that start with an a and ...
 12.1.5: L1 is the set of all strings of as, bs, and cs that contain no cs a...
 12.1.6: L1 is the set of all strings of 0s and 1s that start with a 0, and ...
 12.1.7: In 79, add parentheses to make the order of precedence clear in the...
 12.1.8: In 79, add parentheses to make the order of precedence clear in the...
 12.1.9: In 79, add parentheses to make the order of precedence clear in the...
 12.1.10: In 1012 use the convention about order of precedence to eliminate t...
 12.1.11: In 1012 use the convention about order of precedence to eliminate t...
 12.1.12: In 1012 use the convention about order of precedence to eliminate t...
 12.1.13: In 1315 use set notation to derive the language defined by the give...
 12.1.14: In 1315 use set notation to derive the language defined by the give...
 12.1.15: In 1315 use set notation to derive the language defined by the give...
 12.1.16: In 1618 write five strings that belong to the language defined by t...
 12.1.17: In 1618 write five strings that belong to the language defined by t...
 12.1.18: In 1618 write five strings that belong to the language defined by t...
 12.1.19: In 1921 use words to describe the language defined by the given reg...
 12.1.20: In 1921 use words to describe the language defined by the given reg...
 12.1.21: In 1921 use words to describe the language defined by the given reg...
 12.1.22: In 2224 indicate whether the given strings belong to the language d...
 12.1.23: In 2224 indicate whether the given strings belong to the language d...
 12.1.24: In 2224 indicate whether the given strings belong to the language d...
 12.1.25: The language consisting of all strings of 0s and 1s with an odd num...
 12.1.26: The language consisting of all strings of as and bs in which the th...
 12.1.27: The language consisting of strings of xs and ys in which the elemen...
 12.1.28: Let r,s, and t be regular expressions over = {a, b}. In 2830 determ...
 12.1.29: Let r,s, and t be regular expressions over = {a, b}. In 2830 determ...
 12.1.30: Let r,s, and t be regular expressions over = {a, b}. In 2830 determ...
 12.1.31: In 3139 write a regular expression to define the given set of strin...
 12.1.32: In 3139 write a regular expression to define the given set of strin...
 12.1.33: In 3139 write a regular expression to define the given set of strin...
 12.1.34: In 3139 write a regular expression to define the given set of strin...
 12.1.35: In 3139 write a regular expression to define the given set of strin...
 12.1.36: In 3139 write a regular expression to define the given set of strin...
 12.1.37: In 3139 write a regular expression to define the given set of strin...
 12.1.38: In 3139 write a regular expression to define the given set of strin...
 12.1.39: In 3139 write a regular expression to define the given set of strin...
 12.1.40: Write a regular expression to perform a complete check to determine...
 12.1.41: Write a regular expression to define the set of strings of 0s and 1...
Solutions for Chapter 12.1: Formal Languages and Regular Expressions
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 12.1: Formal Languages and Regular Expressions
Get Full SolutionsDiscrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326. This expansive textbook survival guide covers the following chapters and their solutions. Since 41 problems in chapter 12.1: Formal Languages and Regular Expressions have been answered, more than 52792 students have viewed full stepbystep solutions from this chapter. Chapter 12.1: Formal Languages and Regular Expressions includes 41 full stepbystep solutions. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Column space C (A) =
space of all combinations of the columns of A.

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.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to 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!).

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

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.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

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.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

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

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).