 7.2.32E: a. Is log8 27 = log2 3? Why or why not?b. Is log16 9 = log4 3? Why ...
 7.2.1E: The definition of onetoone is stated in two ways:?x1, x2 ? X, if ...
 7.2.2E: Fill in each blank with the word most or least.a. A function F is o...
 7.2.3E: When asked to state the definition of onetoone, a student replies...
 7.2.4E: Let f: X ? Y be a function. True or false? A sufficient condition f...
 7.2.5E: All but two of the following statements are correct ways to express...
 7.2.6E: Let X = {1, 5, 9} and Y = {3, 4, 7}.a. Define f: X ? Y by specifyin...
 7.2.7E: Let X = {a, b, c, d} and Y = {e, f, g}. Define functions F and G by...
 7.2.8E: Let X = {a, b, c} and Y = {w, x, y, z}. Define functions H and K by...
 7.2.9E: Let X = {1, 2, 3}, Y = {1, 2, 3, 4}, and Z = {1, 2}.a. Define a fun...
 7.2.10E: a. Define f: Z ? Z by the rule f(n)= 2n, for all integers n.(i) Is ...
 7.2.11E: a. Define g: Z ? Z by the rule g(n)= 4n ? 5, for all integers n.(i)...
 7.2.12E: a. Define F: Z ? Z by the rule F(n)= 2 ? 3n, for all integers n.(i)...
 7.2.13E: a. Define H: R ? R by the rule H(x)= x2, for all real numbers x.(i)...
 7.2.14E: Explain the mistake in the following “proof.”Theorem: The function ...
 7.2.15E: In a function f is defined on a set of real numbers. Determine whet...
 7.2.16E: In a function f is defined on a set of real numbers. Determine whet...
 7.2.17E: In a function f is defined on a set of real numbers. Determine whet...
 7.2.18E: In a function f is defined on a set of real numbers. Determine whet...
 7.2.19E: Referring to Example 7.2.3, assume that records with the following ...
 7.2.20E: Define Floor: R ? Z by the formula Floor for all real numbers x.a. ...
 7.2.21E: Let S be the set of all strings of 0’s and 1’s, and define l: S ? Z...
 7.2.22E: Let S be the set of all strings of 0’s and 1’s, and define D: S ? Z...
 7.2.23E: Define as follows: For all A in F(A)= the number of elements in A.a...
 7.2.24E: Let S be the set of all strings of a’s and b’s, and define N: S ? Z...
 7.2.25E: Let S be the set of all strings in a’s and b’s, and define C: S ? S...
 7.2.26E: Define S: Z+ ? Z+ by the rule: For all integers n, S(n) = the sum o...
 7.2.27E: Let D be the set of all finite subsets of positive integers, and de...
 7.2.28E: Define G: R × R ? R × R as follows:G(x, y) = (2y, ?x) for all (x, y...
 7.2.29E: Define H: R × R ? R × R as follows:H(x, y) = (x + 1, 2 ? y) for all...
 7.2.30E: Define J: Q × Q ? R by the rule J(r, s)= r + for all (r, s)? Q × Q....
 7.2.31E: Define F: Z+ × Z+ ? Z+ and G: Z+ × Z+ ? Z+ as follows: For all (n, ...
 7.2.33E: Prove that for all positive real numbers b, x, and y with b ? 1,
 7.2.34E: Prove that for all positive real numbers b, x, and y with b ? 1,log...
 7.2.35E: Prove that for all real numbers a, b, and x with b and x positive a...
 7.2.36E: Exercise use the following definition: If f: R ? R and g: R ? R are...
 7.2.37E: Exercise use the following definition: If f: R ? R and g: R ? R are...
 7.2.38E: Exercise use the following definition: If f: R ? R is a function an...
 7.2.39E: Exercise use the following definition: If f: R ? R is a function an...
 7.2.40E: Suppose F: X ? Y is onetoone.a. Prove that for all subsets A ? X,...
 7.2.41E: Suppose F:X ? Y is onto. Prove that for all subsets B ? Y, F(F1(B)...
 7.2.42E: Let X = {a, b, c, d, e} and Y = {s, t, u, v, w}. In a onetoone co...
 7.2.43E: Let X = {a, b, c, d, e} and Y = {s, t, u, v, w}. In a onetoone co...
 7.2.44E: Indicate which of the functions in the referenced exercise are one...
 7.2.45E: Indicate which of the functions in the referenced exercise are one...
 7.2.46E: Indicate which of the functions in the referenced exercise are one...
 7.2.47E: Indicate which of the functions in the referenced exercise are one...
 7.2.48E: In 44–55 indicate which of the functions in the referenced exercise...
 7.2.49E: In 44–55 indicate which of the functions in the referenced exercise...
 7.2.50E: In 44–55 indicate which of the functions in the referenced exercise...
 7.2.51E: In 44–55 indicate which of the functions in the referenced exercise...
 7.2.52E: Indicate which of the functions in the referenced exercise are one...
 7.2.53E: Indicate which of the functions in the referenced exercise are one...
 7.2.54E: Indicate which of the functions in the referenced exercise are one...
 7.2.55E: Indicate which of the functions in the referenced exercise are one...
 7.2.56E: In Example 7.2.8 a onetoone correspondence was defined from the p...
 7.2.57E: Write a computer algorithm to check whether a function from one fin...
 7.2.58E: Write a computer algorithm to check whether a function from one fin...
Solutions for Chapter 7.2: Discrete Mathematics with Applications 4th Edition
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 7.2
Get Full SolutionsSince 58 problems in chapter 7.2 have been answered, more than 25761 students have viewed full stepbystep solutions from this chapter. Chapter 7.2 includes 58 full stepbystep solutions. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4th. Discrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326. This expansive textbook survival guide covers the following chapters and their solutions.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

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

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).

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

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

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

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

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

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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 .

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.

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.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

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

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

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