 1.5.11E: Let S(x) be the predicate “x is a student,” F(x) the predicate “x i...
 1.5.9E: Let L(x, y) be the statement “x loves y.” where the domain for both...
 1.5.1E: Translate these statements into English, where the domain for each ...
 1.5.2E: Translate these statements into English, where the domain for each ...
 1.5.36E: Express each of these statements using quantifiers. Then form the n...
 1.5.37E: Express each of these statements using quantifiers. Then form the n...
 1.5.38E: Express the negations of these propositions using quantifiers, and ...
 1.5.12E: Let I(x) be the statement “x has an Internet connection” and C(x, y...
 1.5.13E: Let M(x, y) be “x has sent y an email message” and T(x. y) be “x h...
 1.5.3E: Let Q(x, y) be the statement “x has sent an email message to y,” w...
 1.5.14E: Use quantifiers and predicates with more than one variable to expre...
 1.5.4E: Let P(x. y) be the statement “Student x has taken class y,” where t...
 1.5.48E: Show that and where all quantifiers have the same nonempty domain, ...
 1.5.5E: Let W(x. y) mean that student .x has visited website y, where the d...
 1.5.49E: a) Show that is logically equivalent to where all quantifiers have ...
 1.5.39E: Find a counterexample, if possible, to these universally quantified...
 1.5.50E: APut these statements in prenex normal form. [Hint: Use logical equ...
 1.5.40E: Find a counter examples, if possible, to these universally quantifi...
 1.5.41E: Use quantifiers to express the associative law for multiplication o...
 1.5.30E: Rewrite each of these statements so that negations appear only with...
 1.5.6E: Let C(x, y) mean that student x is enrolled in class y, where the d...
 1.5.7E: Let T(x. y) mean that student x likes cuisine y, where the domain f...
 1.5.8E: Let Q(x, y) be the statement “student x has been a contestant on qu...
 1.5.42E: Use quantifiers to express the distributive laws of multiplication ...
 1.5.44E: Use quantifiers and logical connectives to express the fact that a ...
 1.5.43E: Use quantifiers and logical connectives to express the fact that ev...
 1.5.16E: A discrete mathematics class contains 1 mathematics major who is a ...
 1.5.17E: Express each of these system specifications using predicates, quant...
 1.5.52E: Express the quantification introduced in Section 1.4, using univers...
 1.5.51E: Show how to transform an arbitrary statement to a statement in pren...
 1.5.31E: Express the negations of each of these statements so that all negat...
 1.5.32E: Express the negations of each of these statements so that all negat...
 1.5.47E: Show that the two statements and where both quantifiers over the fi...
 1.5.46E: Determine the truth value of the statement if the domain for the va...
 1.5.18E: Express each of these system specifications using predicates, quant...
 1.5.19E: Express each of these statements using mathematical and logical ope...
 1.5.20E: Express each of these statements using predicates, quantifiers, log...
 1.5.21E: Use predicates, quantifiers, logical connectives, and mathematical ...
 1.5.23E: Express each of these mathematical statements using predicates, qua...
 1.5.22E: Use predicates, quantifiers, logical connectives, and mathematical ...
 1.5.24E: Translate each of these nested quantifications into an English stat...
 1.5.25E: Translate each of these nested quantifications into an English stat...
 1.5.26E: Let Q(x. y) be the statement “x + y = x – y.” If the domain for bot...
 1.5.27E: Determine the truth value of each of these statements if the domain...
 1.5.28E: Determine the truth value of each of these statements if the domain...
 1.5.29E: Suppose the domain of the propositional function P(x, y) consists o...
 1.5.34E: Find a common domain for the variables x, y, and z for which the st...
 1.5.33E: Rewrite each of these statements so that negations appear only with...
 1.5.35E: Find a common domain for the variables .x, y, z, and w for which th...
 1.5.15E: Use quantifiers and predicates with more than one variable to expre...
 1.5.10E: Let F(x, y) be the statement “x can fool y,” where the domain consi...
 1.5.45E: Determine the truth value of the statement if the domain for the va...
Solutions for Chapter 1.5: Discrete Mathematics and Its Applications 7th Edition
Full solutions for Discrete Mathematics and Its Applications  7th Edition
ISBN: 9780073383095
Solutions for Chapter 1.5
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Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Column space C (A) =
space of all combinations of the columns of A.

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

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

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.

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

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

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

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

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).

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Solvable system Ax = b.
The right side b is in the column space of A.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

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

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

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·

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