 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 302910 students have viewed full stepbystep solutions from this chapter. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

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

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].

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

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.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

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!).

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

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

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

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

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

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.