 1.1.1.1.1: Which of these sentences are propositions? What are the truth value...
 1.1.1.1.2: Which ofthese are propositions? What are the truth values of those ...
 1.1.1.1.3: What is the negation of each of these propositions? a) Today is Thu...
 1.1.1.1.4: Let p and q be the propositions p : I bought a lottery ticket this ...
 1.1.1.1.5: Let p and q be the propositions "Swimming at the New Jersey shore i...
 1.1.1.1.6: Let p and q be the propositions "The election is decided" and "The ...
 1.1.1.1.7: Let p and q be the propositions p : It is below freezing. q : It is...
 1.1.1.1.8: Let p, q, and r be the propositions p : You have the flu. q : You m...
 1.1.1.1.9: Let p and q be the propositions p : You drive over 65 miles per hou...
 1.1.1.1.10: Let p, q, and r be the propositions p : You get an A on the final e...
 1.1.1.1.11: Let p, q, and r be the propositions p : Grizzly bears have been see...
 1.1.1.1.12: Determine whether these biconditionals are true or false. a) 2 + 2 ...
 1.1.1.1.13: Determine whether each of these conditional statements is true or f...
 1.1.1.1.14: Determine whether each of these conditional statements is true or f...
 1.1.1.1.15: For each of these sentences, determine whether an inclusive or or a...
 1.1.1.1.16: For each of these sentences, determine whether an inclusive or or a...
 1.1.1.1.17: For each of these sentences, state what the sentence means ifthe or...
 1.1.1.1.18: Write each of these statements in the form "if p, then q " in Engli...
 1.1.1.1.19: Write each of these statements in the form "if p, then q " in Engli...
 1.1.1.1.20: Write each of these statements in the form "if p, then q " in Engli...
 1.1.1.1.21: Write each of these propositions in the form "p if and only if q " ...
 1.1.1.1.22: Write each of these propositions in the form "p if and only if q " ...
 1.1.1.1.23: State the converse, contrapositive, and inverse of each of these co...
 1.1.1.1.24: State the converse, contrapositive, and inverse of each of these co...
 1.1.1.1.25: How many rows appear in a truth table for each of these compound pr...
 1.1.1.1.26: How many rows appear in a truth table for each of these compound pr...
 1.1.1.1.27: Construct a truth table for each of these compound propositions. a)...
 1.1.1.1.28: Construct a truth table for each of these compound propositions. a)...
 1.1.1.1.29: Construct a truth table for each of these compound propositions. a)...
 1.1.1.1.30: Construct a truth table for each of these compound propositions. a)...
 1.1.1.1.31: Construct a truth table for each of these compound propositions. a)...
 1.1.1.1.32: Construct a truth table for each of these compound propositions. a)...
 1.1.1.1.33: Construct a truth table for each of these compound propositions. a)...
 1.1.1.1.34: Construct a truth table for p + q) + r) + s.
 1.1.1.1.35: Construct a truth table for (p ++ q) ++ (r ++ s).
 1.1.1.1.36: What is the value of x after each of these statements is encountere...
 1.1.1.1.37: Find the bitwise OR, bitwise AND, and bitwise XOR of each of these ...
 1.1.1.1.38: Evaluate each of these expressions. a) I 1000 /\ (0 101 1 v 1 1011)...
 1.1.1.1.39: The truth value of the negation of a proposition in fuzzy logic is ...
 1.1.1.1.40: The truth value of the conjunction of two propositions in fuzzy log...
 1.1.1.1.41: The truth value of the disjunction of two propositions in fuzzy log...
 1.1.1.1.42: Is the assertion "This statement is false" a proposition?
 1.1.1.1.43: The nth statement in a list of 1 00 statements is "Exactly n of the...
 1.1.1.1.44: An ancient Sicilian legend says that the barber in a remote town wh...
 1.1.1.1.45: Each inhabitant of a remote village always tells the truth or alway...
 1.1.1.1.46: An explorer is captured by a group of cannibals. There are two type...
 1.1.1.1.47: Express these system specifications using the propositions p "The m...
 1.1.1.1.48: Express these system specifications using the propositions p "The u...
 1.1.1.1.49: Are these system specifications consistent? "The system is in multi...
 1.1.1.1.50: Are these system specifications consistent? "Whenever the system so...
 1.1.1.1.51: Are these system specifications consistent? "The router can send pa...
 1.1.1.1.52: Are these system specifications consistent? "If the file system is ...
 1.1.1.1.53: What Boolean search would you use to look for Web pages about beach...
 1.1.1.1.54: What Boolean search would you use to look for Web pages about hikin...
 1.1.1.1.55: A says "At least one of us is a knave" and B says nothing.
 1.1.1.1.56: A says "The two of us are both knights" and B says " A is a knave."
 1.1.1.1.57: A says "I am a knave or B is a knight" and B says nothing
 1.1.1.1.58: Both A and B say "I am a knight."
 1.1.1.1.59: A says "We are both knaves" and B says nothing.
 1.1.1.1.60: The police have three suspects for the murder of Mr. Cooper: Mr. Sm...
 1.1.1.1.61: Steve would like to determine the relative salaries ofthree coworke...
 1.1.1.1.62: Five friends have access to a chat room. Is it possible to determin...
 1.1.1.1.63: A detective has interviewed four witnesses to a crime. From the sto...
 1.1.1.1.64: Four friends have been identified as suspects for an unauthorized a...
 1.1.1.1.65: Solve this famous logic puzzle, attributed to Albert Einstein, and ...
Solutions for Chapter 1.1: The Foundations: Logic and Proofs
Full solutions for Discrete Mathematics and Its Applications  6th Edition
ISBN: 9780073229720
Solutions for Chapter 1.1: The Foundations: Logic and Proofs
Get Full SolutionsDiscrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073229720. Since 65 problems in chapter 1.1: The Foundations: Logic and Proofs have been answered, more than 33813 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 6. Chapter 1.1: The Foundations: Logic and Proofs includes 65 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

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.

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

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

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].

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.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

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.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

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 •

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

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.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

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

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

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

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·