 3.4.1E: Let the following law of algebra be the first statement of an argum...
 3.4.2E: Use universal instantiation or universal modus ponens to fill in va...
 3.4.3E: Use universal instantiation or universal modus ponens to fill in va...
 3.4.4E: Use universal instantiation or universal modus ponens to fill in va...
 3.4.5E: Use universal modus tollens to fill in valid conclusions for the ar...
 3.4.6E: Use universal modus tollens to fill in valid conclusions for the ar...
 3.4.7E: Some of the arguments are valid by universal modus ponens or univer...
 3.4.8E: Some of the arguments are valid by universal modus ponens or univer...
 3.4.9E: Some of the arguments are valid by universal modus ponens or univer...
 3.4.10E: Some of the arguments are valid by universal modus ponens or univer...
 3.4.11E: Some of the arguments are valid by universal modus ponens or univer...
 3.4.12E: Some of the arguments are valid by universal modus ponens or univer...
 3.4.13E: Some of the arguments are valid by universal modus ponens or univer...
 3.4.14E: Some of the arguments are valid by universal modus ponens or univer...
 3.4.15E: Some of the arguments are valid by universal modus ponens or univer...
 3.4.16E: Some of the arguments are valid by universal modus ponens or univer...
 3.4.17E: Some of the arguments are valid by universal modus ponens or univer...
 3.4.18E: Some of the arguments are valid by universal modus ponens or univer...
 3.4.19E: Rewrite the statement “No good cars are cheap” in the form “?x, if ...
 3.4.20E: a. Use a diagram to show that the following argument can have true ...
 3.4.21E: Indicate whether the arguments are valid or invalid. Support your a...
 3.4.22E: Indicate whether the arguments are valid or invalid. Support your a...
 3.4.23E: Indicate whether the arguments are valid or invalid. Support your a...
 3.4.24E: Indicate whether the arguments are valid or invalid. Support your a...
 3.4.25E: Indicate whether the arguments are valid or invalid. Support your a...
 3.4.26E: Indicate whether the arguments are valid or invalid. Support your a...
 3.4.27E: Indicate whether the arguments are valid or invalid. Support your a...
 3.4.28E: In exercise, reorder the premises in each of the arguments to show ...
 3.4.29E: In exercise, reorder the premises in each of the arguments to show ...
 3.4.30E: In exercise, reorder the premises in each of the arguments to show ...
 3.4.31E: In exercise, reorder the premises in each of the arguments to show ...
 3.4.32E: In the exercise, reorder the premises in each of the arguments to s...
 3.4.33E: In a single conclusion follows when all the given premises are take...
 3.4.34E: In a single conclusion follows when all the given premises are take...
 3.4.35E: Derive the validity of universal modus tollens from the validity of...
 3.4.36E: Derive the validity of universal form of part(a) of the elimination...
Solutions for Chapter 3.4: Discrete Mathematics with Applications 4th Edition
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 3.4
Get Full SolutionsChapter 3.4 includes 36 full stepbystep solutions. Since 36 problems in chapter 3.4 have been answered, more than 48025 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4. Discrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326. This expansive textbook survival guide covers the following chapters and their solutions.

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.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

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.

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

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

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

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

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.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

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).

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.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.