 8.5.1: Each of the following is a relation on {0, 1, 2, 3}. Draw directed ...
 8.5.2: Each of the following is a relation on {0, 1, 2, 3}. Draw directed ...
 8.5.3: Let S be the set of all strings of as and bs. Define a relation R o...
 8.5.4: Let S be the set of all strings of as and bs. Define a relation R o...
 8.5.5: Let R be the set of all real numbers and define a relation R on R R...
 8.5.6: Let P be the set of all people who have ever lived and define a rel...
 8.5.7: Let P be the set of all people who have ever lived and define a rel...
 8.5.8: Define a relation R on the set Z of all integers as follows: For al...
 8.5.9: Define a relation R on the set of all real numbers R as follows: Fo...
 8.5.10: Suppose R and S are antisymmetric relations on a set A. Must R S al...
 8.5.11: Let A = {a, b}, and suppose A has the partial order relation R wher...
 8.5.12: Prove Theorem 8.5.1.
 8.5.13: Let A = {a, b}. Describe all partial order relations on A.
 8.5.14: Let A = {a, b, c}. a. Describe all partial order relations on A for...
 8.5.15: Suppose a relation R on a set A is reflexive, symmetric, transitive...
 8.5.16: Consider the divides relation on each of the following sets A. Draw...
 8.5.17: Consider the subset relation on P(S) for each of the following sets...
 8.5.18: Let S = {0, 1} and consider the partial order relation R defined on...
 8.5.19: Let S = {0, 1} and consider the partial order relation R defined on...
 8.5.20: Let S = {0, 1} and consider the partial order relation R defined on...
 8.5.21: Consider the divides relation defined on the set A= {1, 2, 22, 23,....
 8.5.22: In 2229, find all greatest, least, maximal, and minimal elements fo...
 8.5.23: In 2229, find all greatest, least, maximal, and minimal elements fo...
 8.5.24: In 2229, find all greatest, least, maximal, and minimal elements fo...
 8.5.25: In 2229, find all greatest, least, maximal, and minimal elements fo...
 8.5.26: In 2229, find all greatest, least, maximal, and minimal elements fo...
 8.5.27: In 2229, find all greatest, least, maximal, and minimal elements fo...
 8.5.28: In 2229, find all greatest, least, maximal, and minimal elements fo...
 8.5.29: In 2229, find all greatest, least, maximal, and minimal elements fo...
 8.5.30: Each of the following sets is partially ordered with respect to the...
 8.5.31: Let A = {a, b, c, d}, and let R be the relation R = {(a, a), (b, b)...
 8.5.32: Let A = {a, b, c, d}, and let R be the relation R = {(a, a), (b, b)...
 8.5.33: Consider the set A = {12, 24, 48, 3, 9} ordered by the divides rela...
 8.5.34: Suppose that R is a partial order relation on a set A and that B is...
 8.5.35: The set P({w, x, y,z}) is partially ordered with respect to the sub...
 8.5.36: The set A = {2, 4, 3, 6, 12, 18, 24} is partially ordered with resp...
 8.5.37: Find a chain of length 2 for the relation defined in exercise 19
 8.5.38: Prove that a partially ordered set is totally ordered if, and only ...
 8.5.39: Suppose that A is a totally ordered set. Use mathematical induction...
 8.5.40: Prove that a nonempty finite partially ordered set has a. at least ...
 8.5.41: Prove that a finite partially ordered set has a. at most one greate...
 8.5.42: Draw a Hasse diagram for a partially ordered set that has two maxim...
 8.5.43: Draw a Hasse diagram for a partially ordered set that has three max...
 8.5.44: Use the algorithm given in the text to find a topological sorting f...
 8.5.45: Use the algorithm given in the text to find a topological sorting f...
 8.5.46: Use the algorithm given in the text to find a topological sorting f...
 8.5.47: Use the algorithm given in the text to find a topological sorting f...
 8.5.48: Use the algorithm given in the text to find a topological sorting f...
 8.5.49: Refer to the prerequisite structure shown in Figure 8.5.1. a. Find ...
 8.5.50: A set S of jobs can be ordered by writing x y to mean that either x...
 8.5.51: Suppose the tasks described in Example 8.5.12 require the following...
Solutions for Chapter 8.5: Partial Order Relations
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 8.5: Partial Order Relations
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 8.5: Partial Order Relations includes 51 full stepbystep solutions. Discrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326. Since 51 problems in chapter 8.5: Partial Order Relations have been answered, more than 50914 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4.

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.

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.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

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

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.

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.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

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

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.

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).

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.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

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

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.