 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 134355 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.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

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.

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

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

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.

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

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

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

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

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).

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

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

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

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

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·