 4.4.1E: For the values of n and d given, find integers q and r such that n ...
 4.4.2E: For the values of n and d given, find integers q and r such that n ...
 4.4.3E: For the values of n and d given, find integers q and r such that n ...
 4.4.4E: For the values of n and d given, find integers q and r such that n ...
 4.4.5E: For the values of n and d given, find integers q and r such that n ...
 4.4.6E: For the values of n and d given, find integers q and r such that n ...
 4.4.7E: Evaluate the expression.Exercisea. 43 div 9________________b. 43 mod 9
 4.4.8E: Evaluate the expression.Exercisea. 50 div 7________________b. 50 mod 7
 4.4.9E: Evaluate the expression.Exercisea. 28 div 5________________b. 28 mod 5
 4.4.10E: Evaluate the expression.Exercisea. 30 div 2________________b. 30 mod 2
 4.4.11E: Check the correctness of formula for the following values of DayT a...
 4.4.12E: Justify formula for general values of DayT and N.FormulaDayN = (Day...
 4.4.13E: On a Monday a friend says he will meet you again in 30 days. What d...
 4.4.14E: If today is Tuesday, what day of the week will it be 1,000 days fro...
 4.4.15E: January 1, 2000, was a Saturday, and 2000 was a leap year. What day...
 4.4.16E: Suppose d is a positive integer and n is any integer. If d  n, wha...
 4.4.17E: Prove that the product of any two consecutive integers is even.
 4.4.18E: The result of exercise suggests that the second apparent blind alle...
 4.4.19E: Prove that for all integers n, n2 ? n + 3 is odd.
 4.4.20E: Suppose a is an integer. If a mod 7 = 4, what is 5a mod 7? In other...
 4.4.21E: Suppose b is an integer. If b mod 12 = 5, what is 8b mod 12? In oth...
 4.4.22E: Suppose c is an integer. If c mod 15 = 3, what is 10c mod 15? In ot...
 4.4.23E: Prove that for all integers n, if n mod 5 = 3 then n2 mod 5 = 4.
 4.4.24E: Prove that for all integers m and n, if m mod 5 = 2 and n mod 5 = 1...
 4.4.25E: Prove that for all integers a and b, if a mod 7 = 5 and b mod 7 = 6...
 4.4.26E: Prove that a necessary and sufficient condition for a nonnegative i...
 4.4.27E: Show that any integer n can be written in one of the three formsn =...
 4.4.28E: a. Use the quotientremainder theorem with d = 3 to prove that the ...
 4.4.29E: a. Use the quotientremainder theorem with d = 3 to prove that the ...
 4.4.30E: a. Use the quotientremainder theorem with d = 3 to prove that the ...
 4.4.31E: You may use the properties listed in Example.ExampleDeriving Additi...
 4.4.32E: You may use the properties listed in Example.ExampleDeriving Additi...
 4.4.33E: You may use the properties listed in Example.ExampleDeriving Additi...
 4.4.34E: Given any integer n, if n > 3, could n, n + 2, and n + 4 all be pri...
 4.4.35E: Prove each of the statement.ExerciseThe fourth power of any integer...
 4.4.36E: Prove each of the statement.ExerciseThe product of any four consecu...
 4.4.37E: Prove each of the statement.ExerciseThe square of any integer has t...
 4.4.38E: Prove each of the statement.ExerciseFor any integer n, n2 + 5 is no...
 4.4.39E: Prove each of the statement.ExerciseThe sum of any four consecutive...
 4.4.40E: Prove each of the statement.ExerciseFor any integer n, n(n2 ?1)(n +...
 4.4.41E: Prove each of the statement.ExerciseFor all integers m, m2 = 5k, or...
 4.4.42E: Prove each of the statement.ExerciseEvery prime number except 2 and...
 4.4.43E: Prove each of the statement.ExerciseIf n is an odd integer, then n4...
 4.4.44E: Prove each of the statement.ExerciseFor all real numbers x and y, ...
 4.4.45E: Prove each of the statement.ExerciseFor all real numbers r and c wi...
 4.4.46E: Prove each of the statement.ExerciseFor all real numbers r and c wi...
 4.4.47E: A matrix M has 3 rows and 4 columns. The 12 entries in the matrix a...
 4.4.48E: Let M be a matrix with m rows and n columns, and suppose that the e...
 4.4.49E: If m, n, and d are integers, d > 0, and m mod d = n mod d, does it ...
 4.4.50E: If m, n, and d are integers, d > 0, and d  (m ? n), what is the re...
 4.4.51E: If m, n, a, b, and d are integers, d > 0, and m mod d = a and n mod...
 4.4.52E: If m, n, a, b, and d are integers, d > 0, and m mod d = a andn mod ...
 4.4.53E: Prove that if m, d, and k are integers and d > 0, then (m + dk) mod...
Solutions for Chapter 4.4: Discrete Mathematics with Applications 4th Edition
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 4.4
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their 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. Chapter 4.4 includes 53 full stepbystep solutions. Since 53 problems in chapter 4.4 have been answered, more than 92404 students have viewed full stepbystep solutions from this chapter.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

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.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

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].

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.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

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.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

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.

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

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

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

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.

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·