 7.4.1E: When asked what it means to say that set A has the same cardinality...
 7.4.2E: Show that “there are as many squares as there are numbers” by exhib...
 7.4.3E: Let 3Z = {n ? Z  n = 3k, for some integer k}. Prove that Z and 3Z ...
 7.4.4E: Let O be the set of all odd integers. Prove that O has the same car...
 7.4.5E: Let 25Z be the set of all integers that are multiples of 25. Prove ...
 7.4.6E: Check that the formula for F given at the end of Example produces t...
 7.4.7E: Use the functions i and j defined in the paragraph following Exampl...
 7.4.8E: Use the result of exercise to prove that 3Z is countable.ExerciseLe...
 7.4.9E: Show that the set of all nonnegative integers is countable by exhib...
 7.4.10E: Let U = {x ? R  0S and U have the same cardinality.
 7.4.11E: Let V = {x ? R  2S and V have the same cardinality.
 7.4.12E: Let a and b be real numbers with a and suppose that W= {x ? R  a P...
 7.4.13E: Draw the graph of the function f defined by the following formula:F...
 7.4.14E: Define a function g from the set of real numbers to S by the follow...
 7.4.15E: Show that the set of all bit strings (strings of 0’s and 1’s) is co...
 7.4.16E: Show that Q, the set of all rational numbers, is countable.
 7.4.17E: Show that the set Q of all rational numbers is dense along the numb...
 7.4.18E: Must the average of two irrational numbers always be irrational? Pr...
 7.4.19E: Show that the set of all irrational numbers is dense along the numb...
 7.4.20E: Give two examples of functions from Z to Z that are onetoone but ...
 7.4.21E: Give two examples of functions from Z to Z that are onto but not on...
 7.4.22E: Define a function g: Z+ × Z+ ? Z+ by the formula g(m, n) = 2m3n for...
 7.4.23E: a. Explain how to use the following diagram to show that Znonneg × ...
 7.4.24E: Prove that the function H defined analytically in exercise b is a o...
 7.4.25E: Prove that 0.1999 … = 0.2.
 7.4.26E: Prove that any infinite set contains a countably infinite subset.
 7.4.27E: If A is any countably infinite set, B is any set, and g: A ? B is o...
 7.4.28E: Prove that a disjoint union of any finite set and any countably inf...
 7.4.29E: Prove that a union of any two countably infinite sets is countably ...
 7.4.30E: Use the result of exercise to prove that the set of all irrational ...
 7.4.31E: Use the results of exercises 1 and 2 to prove that a union of any t...
 7.4.32E: Prove that Z × Z, the Cartesian product of the set of integers with...
 7.4.33E: Use the results of exercises 3, 4, and 5 to prove the following: If...
 7.4.34E: Let be the set of all subsets of set S, and let T be the set of all...
 7.4.35E: Let S be a set and let be the set of all subsets of S. Show that S ...
 7.4.36E: The SchroederBernstein theorem states the following: If A and B ar...
 7.4.37E: Prove that if A and B are any countably infinite sets, then A × B i...
 7.4.38E: Suppose A1, A2, A3, … is an infinite sequence of countable sets. Re...
Solutions for Chapter 7.4: Discrete Mathematics with Applications 4th Edition
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 7.4
Get Full SolutionsSince 38 problems in chapter 7.4 have been answered, more than 57343 students have viewed full stepbystep solutions from this chapter. Chapter 7.4 includes 38 full stepbystep solutions. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4. 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.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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

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

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].

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

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

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).

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.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

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

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

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