 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: 4th. Discrete Mathematics with Applications was written by Sieva Kozinsky 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 24886 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.

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).

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

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!

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.

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

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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].

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

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].

Iterative method.
A sequence of steps intended to approach the desired solution.

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

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

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.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

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