 4.3.1E: Give a reason for your answer. Assume that all variables represent ...
 4.3.2E: Give a reason for your answer. Assume that all variables represent ...
 4.3.3E: Give a reason for your answer. Assume that all variables represent ...
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 4.3.5E: Give a reason for your answer. Assume that all variables represent ...
 4.3.6E: Give a reason for your answer. Assume that all variables represent ...
 4.3.7E: Give a reason for your answer. Assume that all variables represent ...
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 4.3.13E: Give a reason for your answer. Assume that all variables represent ...
 4.3.14E: Fill in the blanks in the following proof that for all integers a a...
 4.3.15E: Prove statement directly from the definition of divisibility.Exerci...
 4.3.16E: Prove statement directly from the definition of divisibility.Exerci...
 4.3.17E: Consider the following statement: The negative of any multiple of 3...
 4.3.18E: Show that the following statement is false: For all integers a and ...
 4.3.19E: For the statement, determine whether the statement is true or false...
 4.3.20E: For the statement, determine whether the statement is true or false...
 4.3.21E: For the statement, determine whether the statement is true or false...
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 4.3.26E: For the statement, determine whether the statement is true or false...
 4.3.27E: For the statement, determine whether the statement is true or false...
 4.3.28E: For the statement, determine whether the statement is true or false...
 4.3.29E: For the statement, determine whether the statement is true or false...
 4.3.30E: For the statement, determine whether the statement is true or false...
 4.3.31E: For the statement, determine whether the statement is true or false...
 4.3.32E: A fastfood chain has a contest in which a card with numbers on it ...
 4.3.33E: Is it possible to have a combination of nickels, dimes, and quarter...
 4.3.34E: Is it possible to have 50 coins, made up of pennies, dimes, and qua...
 4.3.35E: Two athletes run a circular track at a steady pace so that the firs...
 4.3.36E: It can be shown (see exercises 14) that an integer is divisible by...
 4.3.37E: Use the unique factorization theorem to write the following integer...
 4.3.38E: Suppose that in standard factored form a = where k is a positive in...
 4.3.39E: Suppose that in standard factored form a = where k is a positive in...
 4.3.40E: a. If a and b are integers and 12a = 25b, does 12  b? does 25  a?...
 4.3.41E: How many zeros are at the end of 458 • 885? Explain how you can ans...
 4.3.42E: If n is an integer and n > 1, then n! is the product of n and every...
 4.3.43E: In a certain town of the adult men are married to of the adult wome...
 4.3.44E: Prove that if n is any nonnegative integer whose decimal representa...
 4.3.45E: Prove that if n is any nonnegative integer whose decimal representa...
 4.3.46E: Prove that if the decimal representation of a nonnegative integer n...
 4.3.47E: Observe that Since the sum of the digits of 7524 is divisible by 9,...
 4.3.48E: Prove that for any nonnegative integer n, if the sum of the digits ...
 4.3.49E: Given a positive integer n written in decimal form, the alternating...
Solutions for Chapter 4.3: Discrete Mathematics with Applications 4th Edition
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 4.3
Get Full SolutionsDiscrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326. Chapter 4.3 includes 49 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 49 problems in chapter 4.3 have been answered, more than 45008 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

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.

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

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

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

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

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.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

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.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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.

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

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).

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

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

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

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

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.