 8.1.1: As in Example 8.1.2, the congruence modulo 2 relation E is defined ...
 8.1.2: Prove that for all integers m and n, m n is even if, and only if, b...
 8.1.3: The congruence modulo 3 relation, T , is defined from Z to Z as fol...
 8.1.4: Define a relation P on Z as follows: For all m, n Z, mPn m and n ha...
 8.1.5: Let X = {a, b, c}. Recall that P(X) is the power set of X. Define a...
 8.1.6: Let X = {a, b, c}. Define a relation J on P(X) as follows: For all ...
 8.1.7: Define a relation R on Z as follows: For all integers m and n, mRn ...
 8.1.8: Let A be the set of all strings of as and bs of length 4. Define a ...
 8.1.9: Let A be the set of all strings of 0s, 1s, and 2s of length 4. Defi...
 8.1.10: Let A = {3, 4, 5} and B = {4, 5, 6} and let R be the less than rela...
 8.1.11: Let A = {3, 4, 5} and B = {4, 5, 6} and let S be the divides relati...
 8.1.12: a. Suppose a function F: X Y is onetoone but not onto. Is F1 (the...
 8.1.13: Draw the directed graphs of the relations defined in 1318.
 8.1.14: Draw the directed graphs of the relations defined in 1318.
 8.1.15: Draw the directed graphs of the relations defined in 1318.
 8.1.16: Draw the directed graphs of the relations defined in 1318.
 8.1.17: Draw the directed graphs of the relations defined in 1318.
 8.1.18: Draw the directed graphs of the relations defined in 1318.
 8.1.19: Exercises 1920 refer to unions and intersections of relations. Sinc...
 8.1.20: Exercises 1920 refer to unions and intersections of relations. Sinc...
 8.1.21: Define relations R and S on R as follows: R = {(x, y) R R  x < y} ...
 8.1.22: Define relations R and S on R as follows: R = {(x, y) R R  x 2 + y...
 8.1.23: Define relations R and S on R as follows: R = {(x, y) R R  y = x...
 8.1.24: In Example 8.1.7 the result of the query SELECT PatientID#, Name FR...
Solutions for Chapter 8.1: Relations on Sets
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 8.1: Relations on Sets
Get Full SolutionsDiscrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4. Since 24 problems in chapter 8.1: Relations on Sets have been answered, more than 45392 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 8.1: Relations on Sets includes 24 full stepbystep solutions.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

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

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

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.

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].

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

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

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.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

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).

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

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

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).

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.

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.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

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