 8.3.8.3.1: Represent each ofthese relations on {I, 2, 3 } with a matrix (with ...
 8.3.8.3.2: Represent each of these relations on { l, 2, 3, 4} with a matrix (w...
 8.3.8.3.3: List the ordered pairs in the relations on {I, 2, 3 } corresponding...
 8.3.8.3.4: List the ordered pairs in the relations on {I, 2, 3, 4} correspondi...
 8.3.8.3.5: How can the matrix representing a relation R on a set A be used to ...
 8.3.8.3.6: How can the matrix representing a relation R on a set A be used to ...
 8.3.8.3.7: Determine whether the relations represented by the matrices in Exer...
 8.3.8.3.8: Determine whether the relations represented by the matrices in Exer...
 8.3.8.3.9: How many nonzero entries does the matrix representing the relation ...
 8.3.8.3.10: How many nonzero entries does the matrix representing the relation ...
 8.3.8.3.11: How can the matrix for R, the complement ofthe relation R, be found...
 8.3.8.3.12: How can the matrix for R  I , the inverse of the relation R, be fo...
 8.3.8.3.13: Let R be the relation represented by the matrix 1 1] 1 0 . o 1 Find...
 8.3.8.3.14: Let RI and R2 be relations on a set A represented by the matrices F...
 8.3.8.3.15: Let R be the relation represented by the matrix [0 1 0] MR = 0 0 1 ...
 8.3.8.3.16: Let R be a relation on a set A with n elements. If there are k nonz...
 8.3.8.3.17: Let R be a relation on a set A with n elements. If there are k nonz...
 8.3.8.3.18: Draw the directed graphs representing each of the relations from Ex...
 8.3.8.3.19: Draw the directed graphs representing each of the relations from Ex...
 8.3.8.3.20: Draw the directed graph representing each of the relations from Exe...
 8.3.8.3.21: Draw the directed graph representing each ofthe relations from Exer...
 8.3.8.3.22: Draw the directed graph that represents the relation {(a, a), (a, b...
 8.3.8.3.23: In Exercises 2328 list the ordered pairs in the relations represen...
 8.3.8.3.24: In Exercises 2328 list the ordered pairs in the relations represen...
 8.3.8.3.25: In Exercises 2328 list the ordered pairs in the relations represen...
 8.3.8.3.26: In Exercises 2328 list the ordered pairs in the relations represen...
 8.3.8.3.27: In Exercises 2328 list the ordered pairs in the relations represen...
 8.3.8.3.28: In Exercises 2328 list the ordered pairs in the relations represen...
 8.3.8.3.29: How can the directed graph of a relation R on a finite set A be use...
 8.3.8.3.30: How can the directed graph of a relation R on a finite set A be use...
 8.3.8.3.31: Determine whether the relations represented by the directed graphs ...
 8.3.8.3.32: Determine whether the relations represented by the directed graphs ...
 8.3.8.3.33: Let R be a relation on a set A. Explain how to use the directed gra...
 8.3.8.3.34: Let R be a relation on a set A. Explain how to use the directed gra...
 8.3.8.3.35: Show that ifMR is the matrix representing the relation R, then M l ...
 8.3.8.3.36: Given the directed graphs representing two relations, how can the d...
Solutions for Chapter 8.3: Relations
Full solutions for Discrete Mathematics and Its Applications  6th Edition
ISBN: 9780073229720
Solutions for Chapter 8.3: Relations
Get Full SolutionsDiscrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073229720. Chapter 8.3: Relations includes 36 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 36 problems in chapter 8.3: Relations have been answered, more than 40256 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 6.

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.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

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

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

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

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

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

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 .

Iterative method.
A sequence of steps intended to approach the desired solution.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

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.

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.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

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

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

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.