 1.1.1E: Which of these sentences are propositions? What are the truth value...
 1.1.2E: Which of these are propositions? What are the truth values of those...
 1.1.3E: what is the negation of each of these propositions?a) Mei has an MP...
 1.1.4E: What is the negation of each of these propositions?a) Jennifer and ...
 1.1.5E: What is the negation of each of these propositions?a) Steve has mor...
 1.1.6E: Suppose that Smart phone A has 256 MB RAM end 32 GB ROM, and the re...
 1.1.7E: Suppose that during the most recent fiscal year, the annual revenue...
 1.1.8E: Let p and q be the propositionsp : I bought a lottery ticket this w...
 1.1.9E: Let p and q be the propositions “Swimming at the New Jersey shore i...
 1.1.10E: Let p and q be the propositions “The election is decided” and “The ...
 1.1.11E: Let p and q be the propositionsp : It is below freezing.q : It is s...
 1.1.12E: Let p, q, and r be the propositionsp : You have the flu.q : You mis...
 1.1.13E: Let p and q be the propositionsp : You drive over 65 miles per hour...
 1.1.14E: Let p, q, and r be the propositionsp : You get an A on the final ex...
 1.1.15E: Lei p, q, and r be the propositionsp : Grizzly bears have been seen...
 1.1.16E: Determine whether these biconditionals are true or false.a) 2 + 2= ...
 1.1.17E: Determine whether each of these conditional statements is true or f...
 1.1.18E: Determine whether each of these conditional statements is true or f...
 1.1.19E: For each of these sentences, determine whether an inclusive or, or ...
 1.1.20E: For each of these sentences, determine whether an inclusive or, or ...
 1.1.21E: For each of these sentences, state what the sentence means if the l...
 1.1.22E: Write each of these statements in the form “if p, then q” in Englis...
 1.1.23E: Write each of these statements in the form “if p, then q” in Englis...
 1.1.24E: Write each of these statements in the form “if p, then q” in Englis...
 1.1.25E: Write each of these propositions in the form “p if and only if q” i...
 1.1.26E: Write each of these propositions in the form “p if and only if q” i...
 1.1.27E: State the converse, contrapositive, and inverse of each of these co...
 1.1.28E: State the converse, contrapostive, and inverse of each of these con...
 1.1.29E: How many rows appear in a truth table for each of these compound pr...
 1.1.30E: How many rows appear in a truth table for each of these compound pr...
 1.1.31E: Construct a truth table for each of these compound propositions.
 1.1.32E: Construct a truth table for each of these compound propositions.
 1.1.33E: Construct a truth table for each of these compound propositions.
 1.1.34E: Construct a truth table foreach of these compound propositions.
 1.1.35E: Construct a truth table for each of these compound propositions.
 1.1.36E: Construct a truth table for each of these compound propositions.
 1.1.37E: Construct a truth table for each of these compound propositions.
 1.1.38E: Construct a truth table for
 1.1.39E: Construct a truth table for
 1.1.40E: Explain, without using a truth table, is true when p ,q, and r have...
 1.1.41E: Explain, without using a truth table, why (p ? q ? r ) ? (¬ p ? ¬ q...
 1.1.42E: What is the value of x after each of these statements is encountere...
 1.1.43E: Find the bitwise OR, bitwise AND, and bitwise XOR of each of these ...
 1.1.44E: Evaluate each of these expressions.
 1.1.45E: The truth value of the negation of a proposition in fuzzy logic is ...
 1.1.46E: Fuzzy logic is used in artificial intelligence. In fuzzy logic a pr...
 1.1.47E: The truth value of the disjunction of two propositions in fuzzy log...
 1.1.48E: Is the assertion “This statement is false” a proposition?
 1.1.49E: The nth statement in a list of 100 statements is “Exactly n of the ...
 1.1.50E: An ancient Sicilian legend says that the barber in a remote town wh...
Solutions for Chapter 1.1: Discrete Mathematics and Its Applications 7th Edition
Full solutions for Discrete Mathematics and Its Applications  7th Edition
ISBN: 9780073383095
Solutions for Chapter 1.1
Get Full SolutionsChapter 1.1 includes 50 full stepbystep solutions. Since 50 problems in chapter 1.1 have been answered, more than 150540 students have viewed full stepbystep solutions from this chapter. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 7.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

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!).

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

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

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

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.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

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

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.

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).

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

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!

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.