 1.4.1: All flowers are plants. Pansies are flowers.
 1.4.2: All flowers are plants. Pansies are plants
 1.4.3: All flowers are red or purple. Pansies are flowers. Pansies are not...
 1.4.4: Some flowers are purple. All purple flowers are small.
 1.4.5: Some flowers are red. Some flowers are purple. Pansies are flowers.
 1.4.6: Some flowers are red. Some flowers are purple. Pansies are flowers.
 1.4.7: Justify each step in the following proof sequence of (E x)[P(x) S Q...
 1.4.8: Justify each step in the following proof sequence of (E x)P(x) ` (4...
 1.4.9: Consider the wff (4x)[(E y)P(x, y) ` (E y)Q(x, y)] S (4x)(E y)[P(x,...
 1.4.10: Consider the wff (4y)(E x)Q(x, y) S (E x)(4y)Q(x, y) a. Find an int...
 1.4.11: In Exercises 1116, prove that each wff is a valid argument.
 1.4.12: In Exercises 1116, prove that each wff is a valid argument.
 1.4.13: In Exercises 1116, prove that each wff is a valid argument.
 1.4.14: In Exercises 1116, prove that each wff is a valid argument.
 1.4.15: In Exercises 1116, prove that each wff is a valid argument.
 1.4.16: In Exercises 1116, prove that each wff is a valid argument.
 1.4.17: In Exercises 1730, either prove that the wff is a valid argument or...
 1.4.18: In Exercises 1730, either prove that the wff is a valid argument or...
 1.4.19: In Exercises 1730, either prove that the wff is a valid argument or...
 1.4.20: In Exercises 1730, either prove that the wff is a valid argument or...
 1.4.21: In Exercises 1730, either prove that the wff is a valid argument or...
 1.4.22: In Exercises 1730, either prove that the wff is a valid argument or...
 1.4.23: In Exercises 1730, either prove that the wff is a valid argument or...
 1.4.24: In Exercises 1730, either prove that the wff is a valid argument or...
 1.4.25: In Exercises 1730, either prove that the wff is a valid argument or...
 1.4.26: In Exercises 1730, either prove that the wff is a valid argument or...
 1.4.27: In Exercises 1730, either prove that the wff is a valid argument or...
 1.4.28: In Exercises 1730, either prove that the wff is a valid argument or...
 1.4.29: In Exercises 1730, either prove that the wff is a valid argument or...
 1.4.30: In Exercises 1730, either prove that the wff is a valid argument or...
 1.4.31: The Greek philosopher Aristotle (384322 b.c.e.) studied under Plato...
 1.4.32: Some plants are flowers. All flowers smell sweet. Therefore, some p...
 1.4.33: Every crocodile is bigger than every alligator. Sam is a crocodile....
 1.4.34: There is an astronomer who is not nearsighted. Everyone who wears g...
 1.4.35: Every member of the board comes from industry or government. Everyo...
 1.4.36: There is some movie star who is richer than everyone. Anyone who is...
 1.4.37: Everyone with red hair has freckles. Someone has red hair and big f...
 1.4.38: Cats eat only animals. Something fuzzy exists. Everything thats fuz...
 1.4.39: Every computer science student works harder than somebody, and ever...
 1.4.40: Every ambassador speaks only to diplomats, and some ambassador spea...
 1.4.41: Some elephants are afraid of all mice. Some mice are small. Therefo...
 1.4.42: Every farmer owns a cow. No dentist owns a cow. Therefore no dentis...
 1.4.43: Prove that [(4x)A(x)] 4 (E x)[A(x)] is valid. (Hint: Instead of a p...
 1.4.44: The equivalence of Exercise 43 says that if it is false that every ...
Solutions for Chapter 1.4: Predicate Logic
Full solutions for Mathematical Structures for Computer Science  7th Edition
ISBN: 9781429215107
Solutions for Chapter 1.4: Predicate Logic
Get Full SolutionsSince 44 problems in chapter 1.4: Predicate Logic have been answered, more than 21625 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Mathematical Structures for Computer Science, edition: 7. Mathematical Structures for Computer Science was written by and is associated to the ISBN: 9781429215107. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 1.4: Predicate Logic includes 44 full stepbystep solutions.

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

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

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

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

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.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

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.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

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

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

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).

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

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

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.

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