 8.1.52E: Show that any recurrence relation for the sequence {an} can be writ...
 8.1.53E: Construct the algorithm described in the text after Algorithm 1 for...
 8.1.1E: Use mathematical induction to verify the formula derived in Example...
 8.1.2E: a) Find a recurrence relation for the number of permutations of a s...
 8.1.3E: A vending machine dispensing books of stamps accepts only onedolla...
 8.1.4E: A country uses as currency coins with values of 1 peso, 2 pesos. 5 ...
 8.1.5E: How many ways are there to pay a bill of 17 pesos using the currenc...
 8.1.7E: a) Find a recurrence relation for the number of bit strings of leng...
 8.1.6E: a) Find a recurrence relation for the number of strictly increasing...
 8.1.10E: a) Find a recurrence relation for the number of bit strings of leng...
 8.1.8E: a) Find a recurrence relation for the number of bit strings of leng...
 8.1.9E: a) Find a recurrence relation for the number of bit strings of leng...
 8.1.12E: a) Find a recurrence relation for the number of ways to climb n sta...
 8.1.11E: a) Find a recurrence relation for the number of ways to climb n sta...
 8.1.14E: a) Find a recurrence relation for the number of ternary strings of ...
 8.1.13E: a) Find a recurrence relation for the number of ternary strings of ...
 8.1.16E: a) Find a recurrence relation for the number of ternary strings of ...
 8.1.15E: a) Find a recurrence relation for the number of ternary strings of ...
 8.1.18E: a) Find a recurrence relation for the number of ternary strings of ...
 8.1.17E: a) Find a recurrence relation for the number of ternary strings of ...
 8.1.20E: A bus driver pays all tolls, using only nickels and dimes, by throw...
 8.1.19E: Messages are transmitted over a communications channel using two si...
 8.1.22E: a) Find the recurrence relation satisfied by Rn, where Rn is the nu...
 8.1.21E: a) Find the recurrence relation satisfied by Rn, where Rn is the nu...
 8.1.25E: How many bit sequences of length seven contain an even number of 0s?
 8.1.23E: a) Find the recurrence relation satisfied by Sn, where Sn is the nu...
 8.1.24E: Find a recurrence relation for the number of bit sequences of lengt...
 8.1.26E: a) Find a recurrence relation for the number of ways to completely ...
 8.1.27E: a) Find a recurrence relation for the number of ways to lay out a w...
 8.1.28E: Show that the Fibonacci numbers satisfy the recurrence relation fn ...
 8.1.29E: Let S(m,n) denote the number of onto functions from a set with m el...
 8.1.30E: a) Write out all the ways the product x0? x1? x2? x3? x4 can be par...
 8.1.31E: a) Use the recurrence relation developed in Example 5 to determine ...
 8.1.35E: Show that J(n) satisfies the recurrence relation J(2n) = 2J(n) ? 1 ...
 8.1.32E: In the Tower of Hanoi puzzle, suppose our goal is to transfer all n...
 8.1.34E: Use the values you found in Exercise 33 to conjecture a formula for...
 8.1.33E: Determine the value of J(n) for each integer n with 1 ? n ? 16.
 8.1.36E: Use mathematical induction to prove the formula you conjectured in ...
 8.1.37E: Determine J (100), J (1000), and J (10,000) from your formula for J...
 8.1.38E: Show that the Reve’s puzzle with three disks can be solved using fi...
 8.1.39E: Show that the Reve’s puzzle with four disks can be solved using nin...
 8.1.40E: Describe the moves made by the FrameStewart algorithm, with k chos...
 8.1.41E: Show that if R(n) is the number of moves used by the FrameStewart ...
 8.1.42E: Show that if k is as chosen in Exercise 41, then R(n) ? R(n ? 1) = ...
 8.1.43E: Show that if k is as chosen in Exercise 41. then
 8.1.44E: Use Exercise 43 to give an upper bound on the number of moves requi...
 8.1.46E: Find ?an for the sequence {an}, wherea) an = 4.________________ b) ...
 8.1.45E: Show that R(n) is Let {an} be a sequence of real numbers. The backw...
 8.1.47E: Find ?2an for the sequences in Exercise 46.
 8.1.48E: Show that an?1 = an ? ?an.
 8.1.51E: Express the recurrence relation an = an?k + an?2 in terms of an, ?a...
 8.1.50E: Prove that an?k can be expressed in terms of an, ?an, ?2an ,..., ?kan.
 8.1.49E: Show that
 8.1.54E: Use Algorithm 1 to determine the maximum number of total attendees ...
 8.1.55E: Use Algorithm 1 to determine the maximum number of total attendees ...
 8.1.56E: In this exercise we will develop a dynamic programming algorithm fo...
 8.1.57E: Dynamic programming can be used to develop an algorithm for solving...
Solutions for Chapter 8.1: Discrete Mathematics and Its Applications 7th Edition
Full solutions for Discrete Mathematics and Its Applications  7th Edition
ISBN: 9780073383095
Solutions for Chapter 8.1
Get Full SolutionsSince 57 problems in chapter 8.1 have been answered, more than 200521 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 7. Chapter 8.1 includes 57 full stepbystep solutions.

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).

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

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

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.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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.

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

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

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.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

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.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

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.

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

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.