 6.1.1E: In each of (a)–(f), answer the following questions: Is A ? B? Is B ...
 6.1.2E: Complete the proof from Example: Prove that B ? A whereA = {m ? Z ...
 6.1.3E: Let sets R, S, and T be defined as follows:R = {x ? Z  x is divisi...
 6.1.4E: Let A = {n ? Z  n = 5r for some integer r} and B = {m ? Z  m = 20...
 6.1.5E: Let C = {n ? Z  n = 6r – 5 for some integer r} and D = {m ? Z  m ...
 6.1.6E: Let A = {x ? Z  x = 5a + 2 for some integer a}, B = {y ? Z  y = 1...
 6.1.7E: Let A = {x ? Z  x = 6a + 4 for some integer a}, B = {y ? Z  y = 1...
 6.1.8E: Write in words how to read each of the following out loud. Then wri...
 6.1.9E: Complete the following sentences without using the symbols ?, ?, or...
 6.1.10E: Let A = {1, 3, 5, 7, 9}, B = {3, 6, 9}, and C = {2, 4, 6, 8}. Find ...
 6.1.11E: Let the universal set be the set R of all real numbers and let A = ...
 6.1.12E: Let the universal set be the set R of all real numbers and let A = ...
 6.1.13E: Indicate which of the following relationships are true and which ar...
 6.1.14E: In each of the following, draw a Venn diagram for sets A, B, and C ...
 6.1.15E: Draw Venn diagrams to describe sets A, B, and C that satisfy the gi...
 6.1.16E: Let A = {a, b, c}, B = {b, c, d}, and C = {b, c, e}.a. Find A ? (B ...
 6.1.17E: Consider the Venn diagram shown below. For each of (a)–(f), copy th...
 6.1.18E: a. Is the number 0 in ?Why?________________b. Is = { }? Why?_______...
 6.1.19E: Let Ai = {i, i2} for all integers i = 1, 2, 3, 4.a. A1 ? A2 ? A3 ? ...
 6.1.20E: Let Bi ={x ? R  0 ? x ? i} for all integers i = 1, 2, 3, 4.a. B1 ?...
 6.1.21E: Let Ci = {i, –i} for all nonnegative integers i.a. ________________...
 6.1.22E: Let Di ={x ? R  – i ? x ? i } = [–i, i ] for all nonnegative integ...
 6.1.23E: Let for all positive integers i.a. ________________b. _____________...
 6.1.24E: Let Wi ={x ? R  x > i } = (i, ?) for all nonnegative integers i.a....
 6.1.25E: Let for all positive integers i.a. ________________b. _____________...
 6.1.26E: Let for all positive integers i.a. ________________b. _____________...
 6.1.27E: a. Is {{a, d, e}, {b, c}, {d, f}} a partition of {a, b, c, d, e, f}...
 6.1.28E: Let E be the set of all even integers and O the set of all odd inte...
 6.1.29E: Let R be the set of all real numbers. Is {R+, R–, {0}} a partition ...
 6.1.30E: Let Z be the set of all integers and letA0 = {n ? Z  n = 4k, for s...
 6.1.31E: Suppose A = {1, 2} and B = {2, 3}. Find each of the following:a. __...
 6.1.32E: a. Suppose A = {1} and B = {u, v}. Find .b. Suppose X = {a, b} and ...
 6.1.33E: a. Find .________________b. Find .________________c. Find .
 6.1.34E: Let A1 = {1, 2, 3}, A2 = {u, v}, and A3 = {m, n}. Find each of the ...
 6.1.35E: Let A = {a, b}, B = {1, 2}, and C = {2, 3}. Find each of the follow...
 6.1.36E: Trace the action of Algorithm 6.1.1 on the variables i, j , found, ...
 6.1.37E: Trace the action of Algorithm 6.1.1 on the variables i, j , found, ...
 6.1.38E: Write an algorithm to determine whether a given element x belongs t...
Solutions for Chapter 6.1: Discrete Mathematics with Applications 4th Edition
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 6.1
Get Full SolutionsSince 38 problems in chapter 6.1 have been answered, more than 23942 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4th. Chapter 6.1 includes 38 full stepbystep solutions. Discrete Mathematics with Applications was written by Sieva Kozinsky and is associated to the ISBN: 9780495391326.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

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.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

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 .

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

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.

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

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

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

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

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

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