 9.5.48E: List the ordered pairs in the equivalence relations produced by the...
 9.5.57E: Consider the equivalence relation from Example 2, namely, R = {(x, ...
 9.5.59E: Let R be the relation on the set of all colorings of the 2 × 2 chec...
 9.5.50E: Show that the partition of the set of people living in the United S...
 9.5.1E: Which of these relations on {0, 1, 2, 3} are equivalence relations?...
 9.5.2E: Which of these relations on the set of all people are equivalence r...
 9.5.3E: Which of these rela tions on the set of all functions from Z to Z a...
 9.5.4E: Define three equivalence relations on the set of students in your d...
 9.5.5E: Define three equivalence relations on the set of buildings on a col...
 9.5.6E: Define three equivalence relations on the set of classes offered at...
 9.5.7E: Show that the relation of logical equivalence on the set of all com...
 9.5.8E: Let R be the relation on the set of all sets of real numbers such t...
 9.5.9E: Suppose that A is a nonempty set, and f is a function that has A as...
 9.5.10E: Suppose that A is a nonempty set and R is an equivalence relation o...
 9.5.11E: Show that the relation R consisting of all pairs (x, y) such that x...
 9.5.12E: Show that the relation R consisting of all pairs (x, y) such that x...
 9.5.13E: Show that the relation R consisting of all pairs (x, y) such that x...
 9.5.15E: Let R be the relation on the set of ordered pairs of positive integ...
 9.5.14E: Let R be the relation consisting of all pairs (x, y) such that x an...
 9.5.16E: Let R be the relation on the set of ordered pairs of positive integ...
 9.5.17E: (Requires calculus)a) Show that the relation R on the set of all di...
 9.5.18E: (Requires calculus)a) Let n be a positive integer. Show that the re...
 9.5.19E: Let R be the relation on the set of all URLs (or Web addresses) suc...
 9.5.20E: Let R be the relation on the set of all people who have visited a p...
 9.5.21E: In Exercises 21 determine whether the relation with the directed gr...
 9.5.23E: In Exercises 23 determine whether the relation with the directed gr...
 9.5.22E: In Exercises 22 determine whether the relation with the directed gr...
 9.5.24E: Determine whether the relations represented by these zeroone matri...
 9.5.25E: Show that the relation R on the set of all bit strings such that s ...
 9.5.26E: What are the equivalence classes of the equivalence relations in Ex...
 9.5.27E: What are the equivalence classes of the equivalence relations in Ex...
 9.5.28E: What are the equivalence classes of the equivalence relations in Ex...
 9.5.29E: What is the equivalence class of the bit string 011 for the equival...
 9.5.32E: What are the equivalence classes of the bit strings in Exercise 30 ...
 9.5.30E: What are the equivalence classes of these bit strings for the equiv...
 9.5.31E: What are the equivalence classes of the bit strings in Exercise 30 ...
 9.5.33E: What are the equivalence classes of the bit strings in Exercise 30 ...
 9.5.35E: What is the congruence class [n]5 (that is, the equivalence class o...
 9.5.34E: What are the equivalence classes of the bit strings in Exercise 30 ...
 9.5.36E: What is the congruence class [4]m when m isa) 2?________________b) ...
 9.5.37E: Give a description of each of the congruence classes modulo 6.
 9.5.38E: What is the equivalence class of each of these strings with respect...
 9.5.39E: a) What is the equivalence class of (1, 2) with respect to the equi...
 9.5.40E: a) What is the equivalence class of (1,2) with respect to the equiv...
 9.5.41E: Which of these collections of subsets are partitions of {1,2,3,4,5,...
 9.5.42E: Which of these collections of subsets are partitions of {3,2,1,0...
 9.5.44E: Which of these collections of subsets are partitions of the set of ...
 9.5.43E: Which of these collections of subsets are partitions of the set of ...
 9.5.45E: Which of these are partitions of the set Z × Z of ordered pairs of ...
 9.5.46E: Which of these are partitions of the set of real numbers?a) the neg...
 9.5.47E: List the ordered pairs in the equivalence relations produced by the...
 9.5.49E: Show that the partition formed from congruence classes modulo 6 is ...
 9.5.53E: Show that the partition of the set of all identifiers in C formed b...
 9.5.51E: Show that the partition of the set of bit strings of length 16 form...
 9.5.52E: Show that the partition of the set of all bit strings formed by equ...
 9.5.56E: Suppose that R1 and R2 are equivalence relations on the set S. Dete...
 9.5.55E: Find the smallest equivalence relation on the set {a,b,c,d,e} conta...
 9.5.54E: Suppose that R1 and R2 are equivalence relations on a set A. Let P1...
 9.5.58E: Each bead on a bracelet with three beads is either red, white, or b...
 9.5.60E: a) Let R be the relation on the set of functions from Z+to Z+ such ...
 9.5.61E: Determine the number of different equivalence relations on a set wi...
 9.5.62E: Determine the number of different equivalence relations on a set wi...
 9.5.64E: Do we necessarily get an equivalence relation when we form the symm...
 9.5.65E: Suppose we use Theorem 2 to form a partition P from an equivalence ...
 9.5.63E: Do we necessarily get an equivalence relation when we form the tran...
 9.5.66E: Suppose we use Theorem 2 to form an equivalence relation R from a p...
 9.5.67E: Devise an algorithm to find the smallest equivalence relation conta...
 9.5.68E: Let p(n) denote the number of different equivalence relations on a ...
 9.5.69E: Use Exercise 68 to find the number of different equivalence relatio...
Solutions for Chapter 9.5: Discrete Mathematics and Its Applications 7th Edition
Full solutions for Discrete Mathematics and Its Applications  7th Edition
ISBN: 9780073383095
Solutions for Chapter 9.5
Get Full SolutionsChapter 9.5 includes 69 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 7. Since 69 problems in chapter 9.5 have been answered, more than 186707 students have viewed full stepbystep solutions from this chapter. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

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

Dimension of vector space
dim(V) = number of vectors in any basis for V.

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.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

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

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.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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.

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.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

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

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).