 2.3.1E: Use modus ponens or modus tollens to fill in the blanks in the argu...
 2.3.2E: Use modus ponens or modus tollens to fill in the blanks in the argu...
 2.3.3E: Use modus ponens or modus tollens to fill in the blanks in the argu...
 2.3.4E: Use modus ponens or modus tollens to fill in the blanks in the argu...
 2.3.5E: Use modus ponens or modus tollens to fill in the blanks in the argu...
 2.3.6E: Use truth tables to determine whether the argument forms are valid....
 2.3.7E: Use truth tables to determine whether the argument forms are valid....
 2.3.8E: Use truth tables to determine whether the argument forms are valid....
 2.3.9E: Use truth tables to determine whether the argument forms are valid....
 2.3.10E: pq rp
 2.3.11E: p
 2.3.12E: Use truth tables to show that the following forms of argument are i...
 2.3.13E: Modus tollens:p
 2.3.14E: GeneralizationThe following argument forms are valid:app___________...
 2.3.15E: GeneralizationThe following argument forms are valid:app___________...
 2.3.16E: SpecializationThe following argument forms are valid:a. ___________...
 2.3.17E: SpecializationThe following argument forms are valid:a. ___________...
 2.3.18E: EliminationThe following argument forms are valid:a. ______________...
 2.3.19E: EliminationThe following argument forms are valid:a. pp____________...
 2.3.20E: Example.The following argument form is valid:p ?qq ?r? p ?rMany arg...
 2.3.21E: ExampleProof by Division into CasesThe following argument form is v...
 2.3.22E: Use symbols to write the logical form of each argument in, and then...
 2.3.23E: Use symbols to write the logical form of each argument in, and then...
 2.3.24E: Some of the arguments in are valid, whereas others exhibit the conv...
 2.3.25E: Some of the arguments in are valid, whereas others exhibit the conv...
 2.3.26E: Some of the arguments in are valid, whereas others exhibit the conv...
 2.3.27E: Some of the arguments in are valid, whereas others exhibit the conv...
 2.3.28E: Some of the arguments in are valid, whereas others exhibit the conv...
 2.3.29E: Some of the arguments in are valid, whereas others exhibit the conv...
 2.3.30E: Some of the arguments in are valid, whereas others exhibit the conv...
 2.3.31E: Some of the arguments in are valid, whereas others exhibit the conv...
 2.3.32E: Some of the arguments in are valid, whereas others exhibit the conv...
 2.3.33E: Give an example (other than Example) of a valid argument with a fal...
 2.3.34E: Give an example (other than Example) of an invalid argument with a ...
 2.3.35E: Explain in your own words what distinguishes a valid form of argume...
 2.3.36E: Given the following information about a computer program, find the ...
 2.3.37E: In the back of an old cupboard you discover a note signed by a pira...
 2.3.38E: You are visiting the island described in Example 2.3.14 and have th...
 2.3.39E: The famous detective Percule Hoirot was called in to solve a baffli...
 2.3.40E: Sharky, a leader of the underworld, was killed by one of his own ba...
 2.3.41E: A set of premises and a conclusion are given. Use the valid argumen...
 2.3.42E: A set of premises and a conclusion are given. Use the valid argumen...
 2.3.43E: A set of premises and a conclusion are given. Use the valid argumen...
 2.3.44E: A set of premises and a conclusion are given. Use the valid argumen...
Solutions for Chapter 2.3: Discrete Mathematics with Applications 4th Edition
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 2.3
Get Full SolutionsDiscrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4. Chapter 2.3 includes 44 full stepbystep solutions. Since 44 problems in chapter 2.3 have been answered, more than 47727 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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].

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

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.

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

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

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

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

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

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.

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!

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

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