 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 91457 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.

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

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

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

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.

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

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.

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.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

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

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

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.

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

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

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

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

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

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