- 1.R.1RQ: a) Define the negation of a proposition.________________b) What is ...
- 1.R.2RQ: a) Define (using truth tables) the disjunction, conjunction, exclus...
- 1.R.3RQ: a) d Describe at least five different ways to write the conditional...
- 1.R.4RQ: a) What does it mean for two propositions to be logically equivalen...
- 1.R.5RQ: a) Given a truth table, explain how to use disjunctive normal form ...
- 1.R.6RQ: What arc the universal and existential quantifications of a predica...
- 1.R.7RQ: a) What is the difference between the quantification and where P(x,...
- 1.R.8RQ: Describe what is meant by a valid argument in propositional logic a...
- 1.R.9RQ: Use rules of inference lo show that if the premises "All zebras hav...
- 1.R.10RQ: a) Describe what is meant by a direct proof, a proof by contraposit...
- 1.R.11RQ: a) Describe a way to prove the biconditional p? q.________________b...
- 1.R.12RQ: To prove that the statements p1, p2, p3, and p4 are equivalent, is ...
- 1.R.13RQ: a) Suppose that a statement of the form is false. How can this be p...
- 1.R.14RQ: What is the difference between a constructive and non-constructive ...
- 1.R.15RQ: What arc the elements of a proof that there is a unique element x s...
- 1.R.16RQ: Explain how a proof by cases can be used to prove a result about ab...
Solutions for Chapter 1.R: Discrete Mathematics and Its Applications 7th Edition
Full solutions for Discrete Mathematics and Its Applications | 7th Edition
Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
Invert A by row operations on [A I] to reach [I A-I].
Incidence matrix of a directed graph.
The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .
A symmetric matrix with eigenvalues of both signs (+ and - ).
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = 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 B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
A sequence of steps intended to approach the desired solution.
Jordan form 1 = M- 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.
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).
Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.
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.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
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.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).