 4.2.4E: Convert the binary expansion of each of these integers to a decimal...
 4.2.1E: Convert the decimal expansion of each of these integers to a binary...
 4.2.2E: Convert the decimal expansion of each of these integers to a binary...
 4.2.3E: Convert the binary expansion of each of these integers to a decimal...
 4.2.5E: Convert the octal expansion of each of these integers to a binary e...
 4.2.6E: Convert the binary expansion of each of these integers to an octal ...
 4.2.7E: Convert the hexadecimal expansion of each of these integers to a bi...
 4.2.8E: Convert (BADFACED)16 from its hexadecimal expansion to its binary e...
 4.2.9E: Convert (ABCDEF)16 from its hexadecimal expansion to its binary exp...
 4.2.10E: Convert each of the integers in Exercise 6 from a binary expansion ...
 4.2.11E: Convert (1011 0111 1011)2 from its binary expansion to its hexadeci...
 4.2.12E: Convert (1 1000 0110 0011)2 from its binary expansion to its hexade...
 4.2.13E: Show that the hexadecimal expansion of a positive integer can be ob...
 4.2.14E: Show that the binary expansion of a positive integer can be obtaine...
 4.2.15E: Show that the octal expansion of a positive integer can be obtained...
 4.2.16E: Show that the binary expansion of a positive integer can be obtaine...
 4.2.17E: Convert (7345321)8 to its binary expansion and (10 1011 1011)2 to i...
 4.2.18E: Give a procedure for converting from the hexadecimal expansion of a...
 4.2.19E: Give a procedure for converting from the octal expansion of an inte...
 4.2.20E: Find the sum and the product of each of these pairs of numbers. Exp...
 4.2.21E: Find the sum and the product of each of these pairs of numbers. Exp...
 4.2.22E: Find the sum and product of each of these pairs of numbers. Express...
 4.2.23E: Find the sum and product of each of these pairs of numbers. Express...
 4.2.24E: Find the sum and product of each of these pairs of numbers. Express...
 4.2.25E: Use Algorithm 5 to find 7644 ?mod? 645.
 4.2.26E: Use Algorithm 5 to find 11644 mod 645.
 4.2.27E: Use Algorithm 5 to find 32003 mod 99.
 4.2.28E: Use Algorithm 5 to find 1231001 mod 101.
 4.2.29E: Show that every positive integer can be represented uniquely as the...
 4.2.30E: It can be shown that every integer can be uniquely represented in t...
 4.2.31E: Show that a positive integer is divisible by 3 if and only if the s...
 4.2.32E: Show that a positive integer is divisible by 11 if and only if the ...
 4.2.33E: Show that a positive integer is divisible by 3 if and only if the d...
 4.2.34E: Find the one's complement representations, using bit strings of len...
 4.2.35E: What integer docs each of the following one's complement representa...
 4.2.36E: If m is a positive integer less than 2nl, how is the one's complem...
 4.2.37E: How is the one's complement representation of the sum of two intege...
 4.2.38E: How is the one's complement representation of the difference of two...
 4.2.39E: Show that the integer m with one's complement representation (an1a...
 4.2.40E: Two's complement representations of integers are also used to simpl...
 4.2.41E: Two's complement representations of integers are also used to simpl...
 4.2.42E: Two’s complement representations of integers are also used to simpl...
 4.2.43E: Answer Exercise 37 for two's complement expansions.
 4.2.44E: Answer Exercise 38 for two's complement expansions.
 4.2.45E: Show that the integer m with two's complement representation (an1a...
 4.2.46E: Give a simple algorithm for forming the two's complement representa...
 4.2.47E: Sometimes integers are encoded by using fourdigit binary expansion...
 4.2.48E: Find the Cantor expansions ofa) 2.________________b) 7.____________...
 4.2.49E: Describe an algorithm that finds the Cantor expansion of an integer.
 4.2.50E: Describe an algorithm to add two integers from their Cantor expansi...
 4.2.51E: Add (10111)2 and (11010)2 by working through each step of the algor...
 4.2.52E: Multiply (1110)2 and (1010)2 by working through each step of the al...
 4.2.53E: Describe an algorithm for finding the difference of two binary expa...
 4.2.54E: Estimate the number of bit operations used to subtract two binary e...
 4.2.55E: Devise an algorithm that, given the binary expansions of the intege...
 4.2.56E: How many bit operations does the comparison algorithm from Exercise...
 4.2.57E: Estimate the complexity of Algorithm 1 for finding the base b expan...
 4.2.58E: Show that Algorithm 5 uses O((log m)2 log n) bit operations to find...
 4.2.59E: Show that Algorithm 4 uses O(q log a) bit operations, assuming that...
Solutions for Chapter 4.2: Discrete Mathematics and Its Applications 7th Edition
Full solutions for Discrete Mathematics and Its Applications  7th Edition
ISBN: 9780073383095
Solutions for Chapter 4.2
Get Full SolutionsThis textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 7. Since 59 problems in chapter 4.2 have been answered, more than 441574 students have viewed full stepbystep solutions from this chapter. Chapter 4.2 includes 59 full stepbystep solutions. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095. This expansive textbook survival guide covers the following chapters and their solutions.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

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

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

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

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.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

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

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.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

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

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

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

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·

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.