 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: 4. 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 57343 students have viewed full stepbystep solutions from this chapter.

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).

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Column space C (A) =
space of all combinations of the columns of A.

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

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

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.

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].

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

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

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.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

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

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

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.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!