 8.1.1E: As in Example, the congruence modulo 2 relation E is defined from Z...
 8.1.2E: Prove that for all integers m and n, m –n is even if, and only if, ...
 8.1.3E: The congruence modulo 3 relation, T, is defined from Z to Z as foll...
 8.1.4E: Define a relation P on Z as follows: For all m, n ? Z,m P n ? m and...
 8.1.5E: Let X = {a, b, c}. Recall that is the power set of X. Define a rela...
 8.1.6E: Let X = {a, b, c}. Define a relation J on as follows: For all A, B ...
 8.1.7E: Define a relation R on Z as follows: For all integers m and n,m R n...
 8.1.8E: Let A be the set of all strings of a’s and b’s of length 4. Define ...
 8.1.9E: Let A be the set of all strings of 0’s, 1’s, and 2’s of length 4. D...
 8.1.10E: Let A = {3, 4, 5} and B = {4, 5, 6} and let R be the “less than” re...
 8.1.11E: Let A = {3, 4, 5} and B = {4, 5, 6} and let S be the “divides” rela...
 8.1.12E: a. Suppose a function F: X ? Y is onetoone but not onto. Is F1 (...
 8.1.13E: Draw the directed graphs of the relations.ExerciseDefine a relation...
 8.1.14E: Draw the directed graphs of the relations.ExerciseDefine a relation...
 8.1.15E: Draw the directed graphs of the relations.ExerciseLet A = {2, 3, 4,...
 8.1.16E: Draw the directed graphs of the relations.ExerciseLet A = {5, 6, 7,...
 8.1.17E: Draw the directed graphs of the relations.ExerciseLet A = {2, 3, 4,...
 8.1.18E: Draw the directed graphs of the relations.ExerciseLet A = {0, 1, 2,...
 8.1.19E: Refer to unions and intersections of relations. Since relations are...
 8.1.20E: Refer to unions and intersections of relations. Since relations are...
 8.1.21E: Define relations R and S on R as follows:R = {(x, y) ? R × R  x} a...
 8.1.22E: Define relations R and S on R as follows:R = {(x, y) ? R × R x2 + ...
 8.1.23E: Define relations R and S on R as follows:R = {(x, y) ? R × R y = ...
 8.1.24E: In Example 8.1.7 the result of the query SELECT Patient?ID#, Name F...
Solutions for Chapter 8.1: Discrete Mathematics with Applications 4th Edition
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 8.1
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 8.1 includes 24 full stepbystep solutions. Since 24 problems in chapter 8.1 have been answered, more than 45017 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4. Discrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326.

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).

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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

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

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.

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.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

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.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

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.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as 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.

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.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.