 5.2.1E: Use mathematical induction (and the proof of Proposition as a model...
 5.2.2E: Use mathematical induction to show that any postage of at least 12¢...
 5.2.3E: For each positive integer n, let P(n) be the formula a. Write P(1)....
 5.2.4E: For each integer n with n ? 2, let P (n) be the formula a. Write P ...
 5.2.5E: Fill in the missing pieces in the following proof that1 + 3 + 5 + …...
 5.2.6E: Prove each statement i using mathematical induction. Do not derive ...
 5.2.7E: Prove each statement i using mathematical induction. Do not derive ...
 5.2.8E: Prove each statement i using mathematical induction. Do not derive ...
 5.2.9E: Prove each statement i using mathematical induction. Do not derive ...
 5.2.10E: Prove the statements by mathematical induction.Exercise for all int...
 5.2.11E: Prove the statements by mathematical induction.Exercise for all int...
 5.2.12E: Prove the statements by mathematical induction.Exercise for all int...
 5.2.13E: Prove the statements by mathematical induction.Exercise for all int...
 5.2.14E: for all integers n ? 0.
 5.2.15E: Prove the statements by mathematical induction.Exercise for all int...
 5.2.16E: Prove the statements by mathematical induction.Exercise for all int...
 5.2.17E: Prove the statements by mathematical induction.Exercise for all int...
 5.2.18E: If x is a real number not divisible by n, then for all integers n ? 1,
 5.2.19E: (For students who have studied calculus) Use mathematical induction...
 5.2.20E: Use the formula for the sum of the first n integers and/or the form...
 5.2.21E: Use the formula for the sum of the first n integers and/or the form...
 5.2.22E: Use the formula for the sum of the first n integers and/or the form...
 5.2.23E: Use the formula for the sum of the first n integers and/or the form...
 5.2.24E: Use the formula for the sum of the first n integers and/or the form...
 5.2.25E: Use the formula for the sum of the first n integers and/or the form...
 5.2.26E: Use the formula for the sum of the first n integers and/or the form...
 5.2.27E: Use the formula for the sum of the first n integers and/or the form...
 5.2.28E: Use the formula for the sum of the first n integers and/or the form...
 5.2.29E: Use the formula for the sum of the first n integers and/or the form...
 5.2.30E: Find a formula in n, a, m, and d for the sum (a + md) + (a + (m + 1...
 5.2.31E: Find a formula in a, r, m, and n for the sumarm + arm+1 + arm+2 + …...
 5.2.32E: You have two parents, four grandparents, eight greatgrandparents, ...
 5.2.33E: Find the mistakes in the proof fragments,ExerciseTheorem: For any i...
 5.2.34E: Find the mistakes in the proof fragments,ExerciseTheorem: For any i...
 5.2.35E: Find the mistakes in the proof fragments,ExerciseTheorem: For any i...
 5.2.36E: Use Theorem to prove that if m and n are any positive integers and ...
 5.2.37E: Use Theorem and the result of exercise 10 to prove that if p is any...
Solutions for Chapter 5.2: Discrete Mathematics with Applications 4th Edition
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 5.2
Get Full SolutionsSince 37 problems in chapter 5.2 have been answered, more than 30668 students have viewed full stepbystep solutions from this chapter. Chapter 5.2 includes 37 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.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

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.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

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.

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).

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

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

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

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

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

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

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.