 2.4.1E: Give the output signals for the circuits in 1–4 if the input signal...
 2.4.2E: Give the output signals for the circuits in 1–4 if the input signal...
 2.4.3E: Give the output signals for the circuits in 1–4 if the input signal...
 2.4.4E: Give the output signals for the circuits in 1–4 if the input signal...
 2.4.5E: In 5–8, write an input/output table for the circuit in the referenc...
 2.4.6E: In 5–8, write an input/output table for the circuit in the referenc...
 2.4.7E: In 5–8, write an input/output table for the circuit in the referenc...
 2.4.8E: In 5–8, write an input/output table for the circuit in the referenc...
 2.4.9E: In 9–12, find the Boolean expression that corresponds to the circui...
 2.4.10E: In 9–12, find the Boolean expression that corresponds to the circui...
 2.4.11E: In 9–12, find the Boolean expression that corresponds to the circui...
 2.4.12E: In 9–12, find the Boolean expression that corresponds to the circui...
 2.4.13E: Construct circuits for the Boolean expressions in 13–17.?P ? Q
 2.4.14E: Construct circuits for the Boolean expressions in 13–17.?(P ? Q)
 2.4.15E: Construct circuits for the Boolean expressions in 13–17.P ? (?P ? ?Q)
 2.4.16E: Construct circuits for the Boolean expressions in 13–17.( P ? Q)? ?R
 2.4.17E: Construct circuits for the Boolean expressions in 13–17.( P ? ?Q) ?...
 2.4.18E: For each of the tables in 18–21, construct (a) a Boolean expression...
 2.4.19E: For each of the tables in 18–21, construct (a) a Boolean expression...
 2.4.20E: For each of the tables in 18–21, construct (a) a Boolean expression...
 2.4.21E: For each of the tables in 18–21, construct (a) a Boolean expression...
 2.4.22E: Design a circuit to take input signals P, Q, and R and output a 1 i...
 2.4.23E: Design a circuit to take input signals P, Q, and R and output a 1 i...
 2.4.24E: The lights in a classroom are controlled by two switches: one at th...
 2.4.25E: An alarm system has three different control panels in three differe...
 2.4.26E: Use the properties listed in Theorem 2.1.1 to show that each pair o...
 2.4.27E: Use the properties listed in Theorem 2.1.1 to show that each pair o...
 2.4.28E: Use the properties listed in Theorem 2.1.1 to show that each pair o...
 2.4.29E: Use the properties listed in Theorem 2.1.1 to show that each pair o...
 2.4.30E: For the circuits corresponding to the Boolean expressions in each o...
 2.4.31E: For the circuits corresponding to the Boolean expressions in each o...
 2.4.32E: The Boolean expression for the circuit in Example 2.4.5 is (P ? Q ?...
 2.4.33E: a. Show that for the Sheffer stroke , P ? Q ? (P  Q)  (P  Q).b....
 2.4.34E: Show that the following logical equivalences hold for the Peirce ar...
Solutions for Chapter 2.4: Discrete Mathematics with Applications 4th Edition
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 2.4
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Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

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

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

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

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

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 .

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

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.

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).

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

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

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.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

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.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.