 3.3.1: Let C be the set of cities in the world, let N be the set of nation...
 3.3.2: Let G(x, y) be x 2 > y. Indicate which of the following statements ...
 3.3.3: The following statement is true: nonzero numbers x, a real number y...
 3.3.4: The following statement is true: real numbers x, an integer n such ...
 3.3.5: The statements in exercises 58 refer to the Tarski world given in E...
 3.3.6: The statements in exercises 58 refer to the Tarski world given in E...
 3.3.7: The statements in exercises 58 refer to the Tarski world given in E...
 3.3.8: The statements in exercises 58 refer to the Tarski world given in E...
 3.3.9: Let D = E = {2, 1, 0, 1, 2}. Explain why the following statements a...
 3.3.10: This exercise refers to Example 3.3.3. Determine whether each of th...
 3.3.11: This exercise refers to Example 3.3.3. Determine whether each of th...
 3.3.12: Let D = E = {2, 1, 0, 1, 2}. Write negations for each of the follow...
 3.3.13: In each of 1319, (a) rewrite the statement in English without using...
 3.3.14: In each of 1319, (a) rewrite the statement in English without using...
 3.3.15: In each of 1319, (a) rewrite the statement in English without using...
 3.3.16: In each of 1319, (a) rewrite the statement in English without using...
 3.3.17: In each of 1319, (a) rewrite the statement in English without using...
 3.3.18: In each of 1319, (a) rewrite the statement in English without using...
 3.3.19: In each of 1319, (a) rewrite the statement in English without using...
 3.3.20: Recall that reversing the order of the quantifiers in a statement w...
 3.3.21: For each of the following equations, determine which of the followi...
 3.3.22: In 22 and 23, rewrite each statement without using variables or the...
 3.3.23: In 22 and 23, rewrite each statement without using variables or the...
 3.3.24: Use the laws for negating universal and existential statements to d...
 3.3.25: Each statement in 2528 refers to the Tarski world of Figure 3.3.1. ...
 3.3.26: Each statement in 2528 refers to the Tarski world of Figure 3.3.1. ...
 3.3.27: Each statement in 2528 refers to the Tarski world of Figure 3.3.1. ...
 3.3.28: Each statement in 2528 refers to the Tarski world of Figure 3.3.1. ...
 3.3.29: For each of the statements in 29 and 30, (a) write a new statement ...
 3.3.30: For each of the statements in 29 and 30, (a) write a new statement ...
 3.3.31: Consider the statement Everybody is older than somebody. Rewrite th...
 3.3.32: Consider the statement Somebody is older than everybody. Rewrite th...
 3.3.33: In 3339, (a) rewrite the statement formally using quantifiers and v...
 3.3.34: In 3339, (a) rewrite the statement formally using quantifiers and v...
 3.3.35: In 3339, (a) rewrite the statement formally using quantifiers and v...
 3.3.36: In 3339, (a) rewrite the statement formally using quantifiers and v...
 3.3.37: In 3339, (a) rewrite the statement formally using quantifiers and v...
 3.3.38: In 3339, (a) rewrite the statement formally using quantifiers and v...
 3.3.39: In 3339, (a) rewrite the statement formally using quantifiers and v...
 3.3.40: In informal speech most sentences of the form There is every are in...
 3.3.41: In informal speech most sentences of the form There is every are in...
 3.3.42: Write the negation of the definition of limit of a sequence given i...
 3.3.43: The following is the definition for limxa f (x) = L: For all real n...
 3.3.44: The notation ! stands for the words there exists a unique. Thus, fo...
 3.3.45: Suppose that P(x) is a predicate and D is the domain of x. Rewrite ...
 3.3.46: Suppose that P(x) is a predicate and D is the domain of x. Rewrite ...
 3.3.47: Suppose that P(x) is a predicate and D is the domain of x. Rewrite ...
 3.3.48: Suppose that P(x) is a predicate and D is the domain of x. Rewrite ...
 3.3.49: Suppose that P(x) is a predicate and D is the domain of x. Rewrite ...
 3.3.50: Suppose that P(x) is a predicate and D is the domain of x. Rewrite ...
 3.3.51: Suppose that P(x) is a predicate and D is the domain of x. Rewrite ...
 3.3.52: Suppose that P(x) is a predicate and D is the domain of x. Rewrite ...
 3.3.53: Suppose that P(x) is a predicate and D is the domain of x. Rewrite ...
 3.3.54: Suppose that P(x) is a predicate and D is the domain of x. Rewrite ...
 3.3.55: Let P(x) and Q(x) be predicates and suppose D is the domain of x. I...
 3.3.56: Let P(x) and Q(x) be predicates and suppose D is the domain of x. I...
 3.3.57: Let P(x) and Q(x) be predicates and suppose D is the domain of x. I...
 3.3.58: Let P(x) and Q(x) be predicates and suppose D is the domain of x. I...
 3.3.59: In 5961, find the answers Prolog would give if the following questi...
 3.3.60: In 5961, find the answers Prolog would give if the following questi...
 3.3.61: In 5961, find the answers Prolog would give if the following questi...
Solutions for Chapter 3.3: Statements with Multiple Quantifiers
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 3.3: Statements with Multiple Quantifiers
Get Full SolutionsThis 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. Chapter 3.3: Statements with Multiple Quantifiers includes 61 full stepbystep solutions. Discrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326. Since 61 problems in chapter 3.3: Statements with Multiple Quantifiers have been answered, more than 47987 students have viewed full stepbystep solutions from this chapter.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

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.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

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!).

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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).

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

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.

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).

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

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

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).

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

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