 4.1.1: In 13, use the definitions of even, odd, prime, and composite to ju...
 4.1.2: In 13, use the definitions of even, odd, prime, and composite to ju...
 4.1.3: In 13, use the definitions of even, odd, prime, and composite to ju...
 4.1.4: Prove the statements in 410
 4.1.5: Prove the statements in 411
 4.1.6: Prove the statements in 412
 4.1.7: Prove the statements in 413
 4.1.8: Prove the statements in 414
 4.1.9: Prove the statements in 415
 4.1.10: Prove the statements in 416
 4.1.11: Disprove the statements in 1113 by giving a counterexample.
 4.1.12: Disprove the statements in 1113 by giving a counterexample.
 4.1.13: Disprove the statements in 1113 by giving a counterexample.
 4.1.14: In 1416, determine whether the property is true for all integers, t...
 4.1.15: In 1416, determine whether the property is true for all integers, t...
 4.1.16: In 1416, determine whether the property is true for all integers, t...
 4.1.17: Prove the statements in 17 and 18 by the method of exhaustion.
 4.1.18: Prove the statements in 17 and 18 by the method of exhaustion.
 4.1.19: a. Rewrite the following theorem in three different ways: as , if _...
 4.1.20: Each of the statements in 2023 is true. For each, (a) rewrite the s...
 4.1.21: Each of the statements in 2023 is true. For each, (a) rewrite the s...
 4.1.22: Each of the statements in 2023 is true. For each, (a) rewrite the s...
 4.1.23: Each of the statements in 2023 is true. For each, (a) rewrite the s...
 4.1.24: Prove the statements in 2434. In each case use only the definitions...
 4.1.25: Prove the statements in 2434. In each case use only the definitions...
 4.1.26: Prove the statements in 2434. In each case use only the definitions...
 4.1.27: Prove the statements in 2434. In each case use only the definitions...
 4.1.28: Prove the statements in 2434. In each case use only the definitions...
 4.1.29: Prove the statements in 2434. In each case use only the definitions...
 4.1.30: Prove the statements in 2434. In each case use only the definitions...
 4.1.31: Prove the statements in 2434. In each case use only the definitions...
 4.1.32: Prove the statements in 2434. In each case use only the definitions...
 4.1.33: Prove the statements in 2434. In each case use only the definitions...
 4.1.34: Prove the statements in 2434. In each case use only the definitions...
 4.1.35: Prove that the statements in 3537 are false.
 4.1.36: Prove that the statements in 3537 are false.
 4.1.37: Prove that the statements in 3537 are false.
 4.1.38: Find the mistakes in the proofs shown in 3842. 38. Theorem: For all...
 4.1.39: Find the mistakes in the proofs shown in 3842. 38. Theorem: For all...
 4.1.40: Find the mistakes in the proofs shown in 3842. 38. Theorem: For all...
 4.1.41: Find the mistakes in the proofs shown in 3842. 38. Theorem: For all...
 4.1.42: Find the mistakes in the proofs shown in 3842. 38. Theorem: For all...
 4.1.43: In 4360 determine whether the statement is true or false. Justify y...
 4.1.44: In 4360 determine whether the statement is true or false. Justify y...
 4.1.45: In 4360 determine whether the statement is true or false. Justify y...
 4.1.46: In 4360 determine whether the statement is true or false. Justify y...
 4.1.47: In 4360 determine whether the statement is true or false. Justify y...
 4.1.48: In 4360 determine whether the statement is true or false. Justify y...
 4.1.49: In 4360 determine whether the statement is true or false. Justify y...
 4.1.50: In 4360 determine whether the statement is true or false. Justify y...
 4.1.51: In 4360 determine whether the statement is true or false. Justify y...
 4.1.52: In 4360 determine whether the statement is true or false. Justify y...
 4.1.53: In 4360 determine whether the statement is true or false. Justify y...
 4.1.54: In 4360 determine whether the statement is true or false. Justify y...
 4.1.55: In 4360 determine whether the statement is true or false. Justify y...
 4.1.56: In 4360 determine whether the statement is true or false. Justify y...
 4.1.57: In 4360 determine whether the statement is true or false. Justify y...
 4.1.58: In 4360 determine whether the statement is true or false. Justify y...
 4.1.59: In 4360 determine whether the statement is true or false. Justify y...
 4.1.60: In 4360 determine whether the statement is true or false. Justify y...
 4.1.61: Suppose that integers m and n are perfect squares. Then m + n + 2 m...
 4.1.62: If p is a prime number, must 2p 1 also be prime? Prove or give a co...
 4.1.63: If n is a nonnegative integer, must 22n + 1 be prime? Prove or give...
Solutions for Chapter 4.1: Direct Proof and Counterexample I: Introduction
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 4.1: Direct Proof and Counterexample I: Introduction
Get Full SolutionsSince 63 problems in chapter 4.1: Direct Proof and Counterexample I: Introduction have been answered, more than 24193 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4th. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 4.1: Direct Proof and Counterexample I: Introduction includes 63 full stepbystep solutions. Discrete Mathematics with Applications was written by Sieva Kozinsky and is associated to the ISBN: 9780495391326.

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).

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.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

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].

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.

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.

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

Outer product uv T
= column times row = rank one matrix.

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.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

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

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

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

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

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

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