 2.1.1: In each of 14 represent the common form of each argument using lett...
 2.1.2: In each of 14 represent the common form of each argument using lett...
 2.1.3: In each of 14 represent the common form of each argument using lett...
 2.1.4: In each of 14 represent the common form of each argument using lett...
 2.1.5: Indicate which of the following sentences are statements. a. 1,024 ...
 2.1.6: Write the statements in 69 in symbolic form using the symbols , , a...
 2.1.7: Write the statements in 69 in symbolic form using the symbols , , a...
 2.1.8: Write the statements in 69 in symbolic form using the symbols , , a...
 2.1.9: Write the statements in 69 in symbolic form using the symbols , , a...
 2.1.10: Let p be the statement DATAENDFLAG is off, q the statement ERROR eq...
 2.1.11: Let p be the statement DATAENDFLAG is off, q the statement ERROR eq...
 2.1.12: Write truth tables for the statement forms in 1215.
 2.1.13: Write truth tables for the statement forms in 1215.
 2.1.14: Write truth tables for the statement forms in 1215.
 2.1.15: Write truth tables for the statement forms in 1215.
 2.1.16: Determine whether the statement forms in 1624 are logically equival...
 2.1.17: Determine whether the statement forms in 1624 are logically equival...
 2.1.18: Determine whether the statement forms in 1624 are logically equival...
 2.1.19: Determine whether the statement forms in 1624 are logically equival...
 2.1.20: Determine whether the statement forms in 1624 are logically equival...
 2.1.21: Determine whether the statement forms in 1624 are logically equival...
 2.1.22: Determine whether the statement forms in 1624 are logically equival...
 2.1.23: Determine whether the statement forms in 1624 are logically equival...
 2.1.24: Determine whether the statement forms in 1624 are logically equival...
 2.1.25: Use De Morgans laws to write negations for the statements in 2531
 2.1.26: Use De Morgans laws to write negations for the statements in 2531
 2.1.27: Use De Morgans laws to write negations for the statements in 2531
 2.1.28: Use De Morgans laws to write negations for the statements in 2531
 2.1.29: Use De Morgans laws to write negations for the statements in 2531
 2.1.30: Use De Morgans laws to write negations for the statements in 2531
 2.1.31: Use De Morgans laws to write negations for the statements in 2531
 2.1.32: Assume x is a particular real number and use De Morgans laws to wri...
 2.1.33: Assume x is a particular real number and use De Morgans laws to wri...
 2.1.34: Assume x is a particular real number and use De Morgans laws to wri...
 2.1.35: Assume x is a particular real number and use De Morgans laws to wri...
 2.1.36: Assume x is a particular real number and use De Morgans laws to wri...
 2.1.37: Assume x is a particular real number and use De Morgans laws to wri...
 2.1.38: In 38 and 39, imagine that num_orders and num_instock are particula...
 2.1.39: In 38 and 39, imagine that num_orders and num_instock are particula...
 2.1.40: Use truth tables to establish which of the statement forms in 4043 ...
 2.1.41: Use truth tables to establish which of the statement forms in 4043 ...
 2.1.42: Use truth tables to establish which of the statement forms in 4043 ...
 2.1.43: Use truth tables to establish which of the statement forms in 4043 ...
 2.1.44: In 44 and 45, determine whether the statements in (a) and (b) are l...
 2.1.45: In 44 and 45, determine whether the statements in (a) and (b) are l...
 2.1.46: In Example 2.1.4, the symbol was introduced to denote exclusive or,...
 2.1.47: In logic and in standard English, a double negative is equivalent t...
 2.1.48: In 48 and 49 below, a logical equivalence is derived from Theorem 2...
 2.1.49: In 48 and 49 below, a logical equivalence is derived from Theorem 2...
 2.1.50: Use Theorem 2.1.1 to verify the logical equivalences in 5054. Suppl...
 2.1.51: Use Theorem 2.1.1 to verify the logical equivalences in 5054. Suppl...
 2.1.52: Use Theorem 2.1.1 to verify the logical equivalences in 5054. Suppl...
 2.1.53: Use Theorem 2.1.1 to verify the logical equivalences in 5054. Suppl...
 2.1.54: Use Theorem 2.1.1 to verify the logical equivalences in 5054. Suppl...
Solutions for Chapter 2.1: Logical Form and Logical Equivalence
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 2.1: Logical Form and Logical Equivalence
Get Full SolutionsChapter 2.1: Logical Form and Logical Equivalence includes 54 full stepbystep solutions. Discrete 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: 4th. This expansive textbook survival guide covers the following chapters and their solutions. Since 54 problems in chapter 2.1: Logical Form and Logical Equivalence have been answered, more than 27504 students have viewed full stepbystep solutions from this chapter.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

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.

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].

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

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

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).

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

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).

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

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

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

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

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·