 5.4.1: Suppose a1, a2, a3,... is a sequence defined as follows: a1 = 1, a2...
 5.4.2: Suppose b1, b2, b3,... is a sequence defined as follows: b1 = 4, b2...
 5.4.3: Suppose that c0, c1, c2,... is a sequence defined as follows: c0 = ...
 5.4.4: Suppose that d1, d2, d3,... is a sequence defined as follows: d1 = ...
 5.4.5: Suppose that e0, e1, e2,... is a sequence defined as follows: e0 = ...
 5.4.6: Suppose that f0, f1, f2,... is a sequence defined as follows: f0 = ...
 5.4.7: Suppose that g1, g2, g3,... is a sequence defined as follows: g1 = ...
 5.4.8: Suppose that h0, h1, h2,... is a sequence defined as follows: h0 = ...
 5.4.9: Define a sequence a1, a2, a3,... as follows: a1 = 1, a2 = 3, and ak...
 5.4.10: The problem that was used to introduce ordinary mathematical induct...
 5.4.11: You begin solving a jigsaw puzzle by finding two pieces that match ...
 5.4.12: The sides of a circular track contain a sequence of cans of gasolin...
 5.4.13: Use strong mathematical induction to prove the existence part of th...
 5.4.14: Any product of two or more integers is a result of successive multi...
 5.4.15: Any sum of two or more integers is a result of successive additions...
 5.4.16: Compute 41, 42, 43, 44, 45, 46, 47, and 48. Make a conjecture about...
 5.4.17: Compute 41, 42, 43, 44, 45, 46, 47, and 48. Make a conjecture about...
 5.4.18: Compute 90, 91, 92, 93, 94, and 95. Make a conjecture about the uni...
 5.4.19: Find the mistake in the following proof that purports to show that ...
 5.4.20: Use the wellordering principle for the integers to prove Theorem 4...
 5.4.21: Use the wellordering principle for the integers to prove the exist...
 5.4.22: a. The Archimedean property for the rational numbers states that fo...
 5.4.23: Use the results of exercise 22 and the wellordering principle for ...
 5.4.24: Use the wellordering principle to prove that given any integer n 1...
 5.4.25: Imagine a situation in which eight people, numbered consecutively 1...
 5.4.26: Suppose P(n) is a property such that 1. P(0), P(1), P(2) are all tr...
 5.4.27: Prove that if a statement can be proved by strong mathematical indu...
 5.4.28: Give examples to illustrate the proof of Theorem 5.4
 5.4.29: It is a fact that every integer n 1 can be written in the form cr 3...
 5.4.30: Use mathematical induction to prove the existence part of the quoti...
 5.4.31: Prove that if a statement can be proved by ordinary mathematical in...
 5.4.32: Use the principle of ordinary mathematical induction to prove the w...
Solutions for Chapter 5.4: Strong Mathematical Induction and the WellOrdering Principle for the Integers
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 5.4: Strong Mathematical Induction and the WellOrdering Principle for the Integers
Get Full SolutionsThis 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. Since 32 problems in chapter 5.4: Strong Mathematical Induction and the WellOrdering Principle for the Integers have been answered, more than 45481 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 5.4: Strong Mathematical Induction and the WellOrdering Principle for the Integers includes 32 full stepbystep solutions.

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.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

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.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

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 .

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

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.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

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

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

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.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

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·

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.