 8.3.1E: Suppose that S = {a, b, c, d, e} and R is a relation on S such that...
 8.3.2E: Each of the following partitions of {0, 1, 2, 3, 4} induces a relat...
 8.3.3E: The relation R is an equivalence relation on the set A. Find the di...
 8.3.4E: The relation R is an equivalence relation on the set A. Find the di...
 8.3.5E: The relation R is an equivalence relation on the set A. Find the di...
 8.3.6E: The relation R is an equivalence relation on the set A. Find the di...
 8.3.7E: The relation R is an equivalence relation on the set A. Find the di...
 8.3.8E: The relation R is an equivalence relation on the set A. Find the di...
 8.3.9E: The relation R is an equivalence relation on the set A. Find the di...
 8.3.10E: The relation R is an equivalence relation on the set A. Find the di...
 8.3.11E: The relation R is an equivalence relation on the set A. Find the di...
 8.3.12E: The relation R is an equivalence relation on the set A. Find the di...
 8.3.13E: In each of 3–14, the relation R is an equivalence relation on the s...
 8.3.14E: In each of 3–14, the relation R is an equivalence relation on the s...
 8.3.15E: Determine which of the following congruence relations are true and ...
 8.3.16E: a. Let R be the relation of congruence modulo 3. Which of the follo...
 8.3.17E: a. Prove that for all integers m and n, m = n (mod 3) if, and only ...
 8.3.18E: a. Give an example of two sets that are distinct but not disjoint.b...
 8.3.19E: (1) prove that the relation is an equivalence relation, and (2) des...
 8.3.20E: (1) prove that the relation is an equivalence relation, and (2) des...
 8.3.21E: (1) prove that the relation is an equivalence relation, and (2) des...
 8.3.22E: (1) prove that the relation is an equivalence relation, and (2) des...
 8.3.23E: In 19?31, (1) prove that the relation is an equivalence relation, a...
 8.3.24E: In 19–31, (1) prove that the relation is an equivalence relation, a...
 8.3.25E: (1) prove that the relation is an equivalence relation, and (2) des...
 8.3.26E: (1) prove that the relation is an equivalence relation, and (2) des...
 8.3.27E: (1) prove that the relation is an equivalence relation, and (2) des...
 8.3.28E: (1) prove that the relation is an equivalence relation, and (2) des...
 8.3.29E: (1) prove that the relation is an equivalence relation, and (2) des...
 8.3.30E: (1) prove that the relation is an equivalence relation, and (2) des...
 8.3.31E: (1) prove that the relation is an equivalence relation, and (2) des...
 8.3.32E: Let A be the set of all straight lines in the Cartesian plane. Defi...
 8.3.33E: Let A be the set of points in the rectangle with x and y coordinate...
 8.3.34E: The documentation for the computer language Java recommends that wh...
 8.3.35E: Find an additional representative circuit for the input/output tabl...
 8.3.36E: Let R be an equivalence relation on a set A. Prove each of the stat...
 8.3.37E: Let R be an equivalence relation on a set A. Prove each of the stat...
 8.3.38E: Let R be an equivalence relation on a set A. Prove each of the stat...
 8.3.39E: Let R be an equivalence relation on a set A. Prove each of the stat...
 8.3.40E: Let R be an equivalence relation on a set A. Prove each of the stat...
 8.3.41E: Let R be an equivalence relation on a set A. Prove each of the stat...
 8.3.42E: Let R be the relation defined in Example.a. Prove that R is reflexi...
 8.3.43E: In Example, define operations of addition (+) and multiplication (•...
 8.3.44E: Let A = Z+× Z+. Define a relation R on A as follows: For all (a, b)...
 8.3.45E: The following argument claims to prove that the requirement that an...
 8.3.46E: Let R be a relation on a set A and suppose R is symmetric and trans...
 8.3.47E: Refer to the quote at the beginning of this section to answer the f...
Solutions for Chapter 8.3: Discrete Mathematics with Applications 4th Edition
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 8.3
Get Full SolutionsChapter 8.3 includes 47 full stepbystep solutions. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4th. This expansive textbook survival guide covers the following chapters and their solutions. Since 47 problems in chapter 8.3 have been answered, more than 24378 students have viewed full stepbystep solutions from this chapter. Discrete Mathematics with Applications was written by Sieva Kozinsky and is associated to the ISBN: 9780495391326.

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

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

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

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.

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

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

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.

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).

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

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

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.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

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

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (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.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).