 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 24124 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4th. Discrete Mathematics with Applications was written by Sieva Kozinsky and is associated to the ISBN: 9780495391326.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

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

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

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

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

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.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

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

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

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.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

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·

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

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