 1.1.1: Which of the following sentences are statements? a. The moon is mad...
 1.1.2: What is the truth value of each of the following statements? a. 8 i...
 1.1.3: Given the truth values A true, B false, and C true, what is the tru...
 1.1.4: Given the truth values A false, B true, and C true, what is the tru...
 1.1.5: Rewrite each of the following statements in the form If A, then B. ...
 1.1.6: Rewrite each of the following statements in the form If A, then B. ...
 1.1.7: Common English has many ways to describe logical connectives. Write...
 1.1.8: Common English has many ways to describe logical connectives. Write...
 1.1.9: Several forms of negation are given for each of the following state...
 1.1.10: Several forms of negation are given for each of the following state...
 1.1.11: Write the negation of each statement. a. If the food is good, then ...
 1.1.12: Write the negation of each statement. a. The processor is fast but ...
 1.1.13: Using the letters indicated for the component statements, translate...
 1.1.14: Using the letters indicated for the component statements, translate...
 1.1.15: Let A, B, and C be the following statements: A Roses are red. B Vio...
 1.1.16: Let A, B, and C, and D be the following statements: A The villain i...
 1.1.17: Use A, B, and C as defined in Exercise 15 to translate the followin...
 1.1.18: Use A, B, and C as defined in Exercise 16 to translate the followin...
 1.1.19: Using letters H, K, A for the component statements, translate the f...
 1.1.20: Using letters A, T, E for the component statements, translate the f...
 1.1.21: Using letters F, B, S for the component statements, translate the f...
 1.1.22: Using letters P, C, B, L for the component statements, translate th...
 1.1.23: Construct truth tables for the following wffs. Note any tautologies...
 1.1.24: Construct truth tables for the following wffs. Note any tautologies...
 1.1.25: Verify the equivalences in the list on page 9 by constructing truth...
 1.1.26: Verify by constructing truth tables that the following wffs are tau...
 1.1.27: Prove the following tautologies by starting with the left side and ...
 1.1.28: Prove the following tautologies by starting with the left side and ...
 1.1.29: We mentioned that (A ` B) ~ C cannot be proved equivalent to A ` (B...
 1.1.30: Let P be the wff A S B. Prove or disprove whether P is equivalent t...
 1.1.31: Write a logical expression for a Web search engine to find sites pe...
 1.1.32: Write a logical expression for a Web search engine to find sites pe...
 1.1.33: Write a logical expression for a Web search engine to find sites pe...
 1.1.34: Write a logical expression for a Web search engine to find sites pe...
 1.1.35: Consider the following pseudocode. repeat i = 1 read a value for x ...
 1.1.36: Suppose that A, B, and C represent conditions that will be true or ...
 1.1.37: Rewrite the following statement form with a simplified conditional ...
 1.1.38: You want your program to execute statement 1 when A is false, B is ...
 1.1.39: Verify that A S B is equivalent to A ~ B.
 1.1.40: a. Using Exercise 39 and other equivalences, prove that the negatio...
 1.1.41: Use algorithm TautologyTest to prove that the following expressions...
 1.1.42: Use algorithm TautologyTest to prove that the following expressions...
 1.1.43: A memory chip from a digital camera has 25 bistable (ONOFF) memory...
 1.1.44: In each case, construct compound wffs P and Q so that the given sta...
 1.1.45: From the truth table for A ~ B, the value of A ~ B is true if A is ...
 1.1.46: Prove that A ! B 4 (A 4 B) is a tautology. Explain why this makes s...
 1.1.47: Every compound statement is equivalent to a statement using only th...
 1.1.48: Show that every compound wff is equivalent to a wff using only the ...
 1.1.49: Show that every compound wff is equivalent to a wff using only the ...
 1.1.50: Prove that there are compound statements that are not equivalent to...
 1.1.51: The binary connective 0 is called the Sheffer stroke, named for the...
 1.1.52: The binary connective T is called the Peirce arrow, named for Ameri...
 1.1.53: Propositional wffs and truth tables belong to a system of twovalue...
 1.1.54: Propositional wffs and truth tables belong to a system of twovalue...
 1.1.55: In a threevalued logic system as described in Exercise 53, how man...
 1.1.56: In 2003, then U.S. Secretary of Defense Donald Rumsfeld won Britain...
 1.1.57: Four machines, A, B, C, and D, are connected on a computer network....
 1.1.58: The Dillies have five teenaged children, two boys named Ollie and R...
 1.1.59: An advertisement for a restaurant at an exclusive club in Honolulu ...
 1.1.60: The following newspaper headline was printed during a murder trial:...
 1.1.61: You meet two of the inhabitants of this country, Percival and Llewe...
 1.1.62: Traveling on, you meet Merlin and Meredith. Merlin says, If I am a ...
 1.1.63: Next, you meet Rothwold and Grymlin. Rothwold says, Either I am a l...
 1.1.64: Finally, you meet Gwendolyn and Merrilaine. Gwendolin says, I am a ...
Solutions for Chapter 1.1: Statements, Symbolic Representation, and Tautologies
Full solutions for Mathematical Structures for Computer Science  7th Edition
ISBN: 9781429215107
Solutions for Chapter 1.1: Statements, Symbolic Representation, and Tautologies
Get Full SolutionsSince 64 problems in chapter 1.1: Statements, Symbolic Representation, and Tautologies have been answered, more than 18921 students have viewed full stepbystep solutions from this chapter. 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. This textbook survival guide was created for the textbook: Mathematical Structures for Computer Science, edition: 7. Chapter 1.1: Statements, Symbolic Representation, and Tautologies includes 64 full stepbystep solutions.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

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

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 .

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

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

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

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.

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 ().

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!

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

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

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

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.