 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 43877 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4. 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 and is associated to the ISBN: 9780495391326.

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

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Column space C (A) =
space of all combinations of the columns of A.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

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

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

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.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

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.

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

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

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).