 2.2.1: Which of the following sentences are statements? For those that are...
 2.2.2: Consider the sets A, B,C and D below. Which of the following statem...
 2.2.3: Which of the following statements are true? Give an explanation for...
 2.2.4: Consider the open sentence P(x) : x(x 1) = 6 over the domain R.(a) ...
 2.2.5: For the open sentence P(x):3x 2 > 4 over the domain Z, determine:(a...
 2.2.6: For the open sentence P(A) : A {1, 2, 3} over the domain S = P({1, ...
 2.2.7: Let P(n): n and n + 2 are primes. be an open sentence over the doma...
 2.2.8: Let P(n) : n2+5n+62 is even.(a) Find a set S1 of three integers suc...
 2.2.9: Find an open sentence P(n) over the domain S = {3, 5, 7, 9} such th...
 2.2.10: Find two open sentences P(n) and Q(n), both over the domain S = {2,...
 2.2.11: State the negation of each of the following statements.(a) 2 is a r...
 2.2.12: Complete the truth table in Figure 2.16.
 2.2.13: State the negation of each of the following statements.(a) The real...
 2.2.14: State the negation of each of the following statements.(a) At least...
 2.2.15: Complete the truth table in Figure 2.17.
 2.2.16: For the sets A = {1, 2, , 10} and B = {2, 4, 6, 9, 12, 25}, conside...
 2.2.17: Let P: 15 is odd. and Q : 21 is prime. State each of the following ...
 2.2.18: Let S = {1, 2,..., 6} and letP(A) : A {2, 4, 6}=. and Q(A) : A = .b...
 2.2.19: Consider the statements P: 17 is even. and Q: 19 is prime. Write ea...
 2.2.20: For statements P and Q, construct a truth table for (P Q) ( P)
 2.2.21: Consider the statements P :2 is rational. and Q : 22/7 is rational....
 2.2.22: Consider the statements:P:2 is rational. Q: 23 is rational. R:3 is ...
 2.2.23: Suppose that {S1, S2} is a partition of a set S and x S. Which of t...
 2.2.24: Two sets A and B are nonempty disjoint subsets of a set S. If x S, ...
 2.2.25: A college student makes the following statement:If I receive an A i...
 2.2.26: A college student makes the following statement:If I dont see my ad...
 2.2.27: The instructor of a computer science class announces to her class t...
 2.2.28: Consider the statement (implication):If Bill takes Sam to the conce...
 2.2.29: Let P and Q be statements. Which of the following implies that P Q ...
 2.2.30: Consider the open sentences P(n):5n + 3 is prime. and Q(n):7n + 1 i...
 2.2.31: In each of the following, two open sentences P(x) and Q(x) over a d...
 2.2.32: In each of the following, two open sentences P(x) and Q(x) over a d...
 2.2.33: In each of the following, two open sentences P(x, y) and Q(x, y) ar...
 2.2.34: Each of the following describes an implication. Write the implicati...
 2.2.35: Let P : 18 is odd. and Q : 25 is even. State P Q in words. Is P Q t...
 2.2.36: Let P(x) : x is odd. and Q(x) : x 2 is odd. be open sentences over ...
 2.2.37: For the open sentences P(x) : x 3 < 1. and Q(x) : x (2, 4). over ...
 2.2.38: Consider the open sentences:P(x) : x = 2. and Q(x) : x 2 = 4.over t...
 2.2.39: For the following open sentences P(x) and Q(x) over a domain S, det...
 2.2.40: In each of the following, two open sentences P(x, y) and Q(x, y) ar...
 2.2.41: Determine all values of n in the domain S = {1, 2, 3} for which the...
 2.2.42: Determine all values of n in the domain S = {2, 3, 4} for which the...
 2.2.43: Let S = {1, 2, 3}. Consider the following open sentences over the d...
 2.2.44: Let S = {1, 2, 3, 4}. Consider the following open sentences over th...
 2.2.45: Let P(n): 2n 1 is a prime. and Q(n): n is a prime. be open sentence...
 2.2.46: For statements P and Q, show that P (P Q) is a tautology.
 2.2.47: For statements P and Q, show that (P ( Q)) (P Q) is a contradiction.
 2.2.48: For statements P and Q, show that (P (P Q)) Q is a tautology. Then ...
 2.2.49: For statements P, Q and R, show that ((P Q) (Q R)) (P R) is a tauto...
 2.2.50: Let R and S be compound statements involving the same component sta...
 2.2.51: For statements P and Q, the implication (P) (Q) is called the inver...
 2.2.52: Let P and Q be statements.(a) Is (P Q) logically equivalent to (P) ...
 2.2.53: For statements P, Q and R, use a truth table to show that each of t...
 2.2.54: For statements P and Q, show that (Q) (P (P)) and Q are logically e...
 2.2.55: For statements P, Q and R, show that (P Q) R and (P R) (Q R) are lo...
 2.2.56: Two compound statements S and T are composed of the same component ...
 2.2.57: Five compound statements S1, S2, S3, S4 and S5 are all composed of ...
 2.2.58: Verify the following laws stated in Theorem 2.18:(a) Let P, Q and R...
 2.2.59: Write negations of the following open sentences:(a) Either x = 0 or...
 2.2.60: Consider the implication: If x and y are even, then x y is even.(a)...
 2.2.61: For a real number x, let P(x) : x 2 = 2. and Q(x) : x = 2. State th...
 2.2.62: Let P and Q be statements. Show that [(P Q) (P Q)] (P Q).
 2.2.63: Let n Z. For which implication is its negation the following?The in...
 2.2.64: For which biconditional is its negation the following?n3 and 7n + 2...
 2.2.65: Let S denote the set of odd integers and letP(x) : x 2 + 1 is even....
 2.2.66: Define an open sentence R(x) over some domain S and then state x S,...
 2.2.67: State the negations of the following quantified statements, where a...
 2.2.68: State the negations of the following quantified statements:(a) For ...
 2.2.69: Let P(n): (5n 6)/3 is an integer. be an open sentence over the doma...
 2.2.70: Determine the truth value of each of the following statements.(a) x...
 2.2.71: The statementFor every integer m, either m 1 or m2 4.can be express...
 2.2.72: Let P(x) and Q(x) be open sentences where the domain of the variabl...
 2.2.73: Let P(x) and Q(x) be open sentences where the domain of the variabl...
 2.2.74: Consider the open sentenceP(x, y,z): (x 1)2 + (y 2)2 + (z 2)2 > 0.w...
 2.2.75: Consider quantified statementFor every s S and t S, st 2 is prime.w...
 2.2.76: Let A be the set of circles in the plane with center (0, 0) and let...
 2.2.77: For a triangle T , let r(T ) denote the ratio of the length of the ...
 2.2.78: Consider the open sentence P(a, b): a/b < 1. where the domain of a ...
 2.2.79: Consider the open sentence Q(a, b): a b < 0. where the domain of a ...
 2.2.80: Give a definition of each of the following and then state a charact...
 2.2.81: Define an integer n to be odd if n is not even. State a characteriz...
 2.2.82: Define a triangle to be isosceles if it has two equal sides. Which ...
 2.2.83: By definition, a right triangle is a triangle one of whose angles i...
 2.2.84: Two distinct lines in the plane are defined to be parallel if they ...
 2.2.85: Construct a truth table for P (Q ( P)).
 2.2.86: Given that the implication (Q R) ( P) is false and Q is false, dete...
 2.2.87: Find a compound statement involving the component statements P and ...
 2.2.88: Determine the truth value of each of the following quantified state...
 2.2.89: Rewrite each of the implications below using (1) only if and (2) su...
 2.2.90: Let P(n): n2 n + 5 is a prime. be an open sentence over a domain S....
 2.2.91: (a) For statements P, Q and R, show that((P Q) R) ((P (R)) (Q)).(b)...
 2.2.92: For a fixed integer n, use Exercise 2.91 to restate the following i...
 2.2.93: For fixed integers m and n, use Exercise 2.91 to restate the follow...
 2.2.94: For a realvalued function f and a real number x, use Exercise 2.91...
 2.2.95: For the set S = {1, 2, 3}, give an example of three open sentences ...
 2.2.96: Do there exist a set S of cardinality 2 and a set {P(n), Q(n), R(n)...
 2.2.97: Let A = {1, 2,..., 6} and B = {1, 2,..., 7}. For x A, let P(x):7x +...
 2.2.98: Let P(x, y,z) be an open sentence, where the domains of x, y and z ...
 2.2.99: Let P(x, y,z) be an open sentence, where the domains of x, y and z ...
 2.2.100: Write each of the following using if, then.(a) A sufficient conditi...
Solutions for Chapter 2: Logic
Full solutions for Mathematical Proofs: A Transition to Advanced Mathematics  3rd Edition
ISBN: 9780321797094
Solutions for Chapter 2: Logic
Get Full SolutionsSince 100 problems in chapter 2: Logic have been answered, more than 5712 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Mathematical Proofs: A Transition to Advanced Mathematics, edition: 3. This expansive textbook survival guide covers the following chapters and their solutions. Mathematical Proofs: A Transition to Advanced Mathematics was written by and is associated to the ISBN: 9780321797094. Chapter 2: Logic includes 100 full stepbystep solutions.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

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

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

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

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

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.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

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!

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

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

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

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

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