 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 48551 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: 4. Chapter 6.1 includes 38 full stepbystep solutions. Discrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

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.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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

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.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

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.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

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

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

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

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.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·