 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 165075 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.

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.

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

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

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.

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.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

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.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

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).

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

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.

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.

Solvable system Ax = b.
The right side b is in the column space of A.

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.

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.