 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: 4. This expansive textbook survival guide covers the following chapters and their solutions. Since 47 problems in chapter 8.3 have been answered, more than 45048 students have viewed full stepbystep solutions from this chapter. Discrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

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!

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

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

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

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

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].

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

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

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

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).

Outer product uv T
= column times row = rank one matrix.

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

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

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.

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.