 3.1.1: A menagerie consists of seven brown dogs, two black dogs, six gray ...
 3.1.2: Indicate which of the following statements are true and which are f...
 3.1.3: Indicate which of the following statements are true and which are f...
 3.1.4: Let Q(n) be the predicate n2 30. a. Write Q(2), Q(2), Q(7), and Q(7...
 3.1.5: Let Q(x, y) be the predicate If x < y then x 2 < y2 with domain for...
 3.1.6: Let R(m, n) be the predicate If m is a factor of n2 then m is a fac...
 3.1.7: Let R(m, n) be the predicate If m is a factor of n2 then m is a fac...
 3.1.8: Let B(x) be 10 < x < 10. Find the truth set of B(x) for each of the...
 3.1.9: Find counterexamples to show that the statements in 912 are false
 3.1.10: Find counterexamples to show that the statements in 912 are false
 3.1.11: Find counterexamples to show that the statements in 912 are false
 3.1.12: Find counterexamples to show that the statements in 912 are false
 3.1.13: Consider the following statement: basketball players x, x is tall. ...
 3.1.14: Consider the following statement: x R such that x 2 = 2. Which of t...
 3.1.15: Rewrite the following statements informally in at least two differe...
 3.1.16: Rewrite each of the following statements in the form x, . a. All di...
 3.1.17: Rewrite each of the following in the form x such that . a. Some exe...
 3.1.18: Let D be the set of all students at your school, and let M(s) be s ...
 3.1.19: Consider the following statement: integers n, if n2 is even then n ...
 3.1.20: Rewrite the following statement informally in at least two differen...
 3.1.21: Rewrite the following statements so that the quantifier trails the ...
 3.1.22: Rewrite each of the following statements in the form x, if then . a...
 3.1.23: Rewrite each of the following statements in the two forms x, if the...
 3.1.24: Rewrite the following statements in the two forms x such that and x...
 3.1.25: The statement The square of any rational number is rational can be ...
 3.1.26: Consider the statement All integers are rational numbers but some r...
 3.1.27: Refer to the picture of Tarskis world given in Example 3.1.13. Let ...
 3.1.28: In 2830, rewrite each statement without using quantifiers or variab...
 3.1.29: In 2830, rewrite each statement without using quantifiers or variab...
 3.1.30: In 2830, rewrite each statement without using quantifiers or variab...
 3.1.31: In any mathematics or computer science text other than this book, f...
 3.1.32: Let R be the domain of the predicate variable x. Which of the follo...
 3.1.33: Let R be the domain of the predicate variables a, b, c, and d. Whic...
Solutions for Chapter 3.1: Predicates and Quantified Statements I
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 3.1: Predicates and Quantified Statements I
Get Full SolutionsChapter 3.1: Predicates and Quantified Statements I includes 33 full stepbystep solutions. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4. This expansive textbook survival guide covers the following chapters and their solutions. Discrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326. Since 33 problems in chapter 3.1: Predicates and Quantified Statements I have been answered, more than 56801 students have viewed full stepbystep solutions from this chapter.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

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.

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.

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

Iterative method.
A sequence of steps intended to approach the desired solution.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

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.

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).

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

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.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

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!

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·

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).