 1.2.43E: Construct a combinatorial circuit using inverters. OR gates, and AN...
 1.2.1E: Translate the given statement into propositional logic using the pr...
 1.2.2E: Translate the given statement into propositional logic using the pr...
 1.2.3E: Translate the given statement into propositional logic using the pr...
 1.2.4E: Translate the given statement into propositional logic using the pr...
 1.2.5E: Translate the given statement into propositional logic using the pr...
 1.2.6E: Translate the given statement into propositional logic using the pr...
 1.2.7E: Express these system specifications using the propositions p “The m...
 1.2.8E: Express these system specifications using the propositions p “The u...
 1.2.9E: Are these system specifications consistent? “The system is in multi...
 1.2.10E: Are these system specifications consistent? “Whenever the system so...
 1.2.11E: Are these system specifications consistent? 'The router can send pa...
 1.2.12E: Are these system specifications consistent? “If the file system is ...
 1.2.13E: What Boolean search would you use to look for Web pages about beach...
 1.2.14E: What Boolean search would you use lo look for Web pages about hikin...
 1.2.15E: Each inhabitant of a remote village always tells the truth or alway...
 1.2.16E: An explorer is captured by a group of cannibals. There are two type...
 1.2.17E: When three professors are seated in a restaurant, the hostess asks ...
 1.2.18E: When planning a party you want to know whom to invite. Among the pe...
 1.2.19E: Relate to inhabitants of the island of knights and knaves created b...
 1.2.20E: Relate to inhabitants of the island of knights and knaves created b...
 1.2.21E: Relate to inhabitants of the island of knights and knaves created b...
 1.2.22E: Relate to inhabitants of the island of knights and knaves created b...
 1.2.23E: Relate to inhabitants of the island of knights and knaves created b...
 1.2.24E: The exercise relates to inhabitants of an island on which there are...
 1.2.25E: The exercise relates to inhabitants of an island on which there are...
 1.2.26E: The exercise relates to inhabitants of an island on which there are...
 1.2.27E: The exercise relates to inhabitants of an island on which there are...
 1.2.28E: The exercise relates to inhabitants of an island on which there are...
 1.2.29E: The exercise relates to inhabitants of an island on which there are...
 1.2.30E: The exercise relates to inhabitants of an island on which there are...
 1.2.31E: The exercise relates to inhabitants of an island on which there are...
 1.2.32E: The exercise is a puzzle that can be solved by translating Statemen...
 1.2.33E: The exercise is a puzzle that can be solved by translating Statemen...
 1.2.34E: The exercise is a puzzle that can be solved by translating Statemen...
 1.2.35E: The exercise is a puzzle that can be solved by translating Statemen...
 1.2.36E: The exercise is a puzzle that can be solved by translating Statemen...
 1.2.37E: The exercise is a puzzle that can be solved by translating Statemen...
 1.2.38E: The exercise is a puzzle that can be solved by translating Statemen...
 1.2.39E: Freedonia has fifty senators. Each senator is either honest or corr...
 1.2.40E: Find the output of each of these combinatorial circuits.
 1.2.41E: Find the output of each of these combinatorial circuits.
 1.2.42E: Construct a combinatorial circuit using inverters, OR gates, and AN...
Solutions for Chapter 1.2: Discrete Mathematics and Its Applications 7th Edition
Full solutions for Discrete Mathematics and Its Applications  7th Edition
ISBN: 9780073383095
Solutions for Chapter 1.2
Get Full SolutionsChapter 1.2 includes 43 full stepbystep solutions. 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. Since 43 problems in chapter 1.2 have been answered, more than 133251 students have viewed full stepbystep solutions from this chapter. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095.

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

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

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

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

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

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.

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.

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.

Iterative method.
A sequence of steps intended to approach the desired solution.

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.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

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.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

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