- 1.2.1: (a) Write a table that gives the different assignments of truth val...
- 1.2.2: State the negation of each of the following statements. (a) 3 > 1. ...
- 1.2.3: For an integer n, consider the following open sentences P(n): 2n > ...
- 1.2.4: Consider the following two statements P: 23 + 32 is even. Q: 4 3 2 ...
- 1.2.5: For an integer n, consider the following two open sentences P(n): n...
- 1.2.6: For an integer n, consider the following two open sentences P(n): 2...
- 1.2.7: For an integer n, consider the following two open sentences P(n) : ...
- 1.2.8: Let P and Q be statements. Construct a truth table for each of the ...
- 1.2.9: Consider the two statements P: 15 is odd. Q : 21 is even. State eac...
- 1.2.10: For two statements P and Q, verify that P Q Q P. 1
- 1.2.11: Verify the following De Morgans Law by a truth table. For two state...
- 1.2.12: Use De Morgans Laws to state the negations of the following. (a) Ei...
- 1.2.13: Use De Morgans Laws to state the negations of the following. (a) Ei...
- 1.2.14: For two statements P and Q, use truth tables to verify the followin...
- 1.2.15: For two nonzero integers a and b such that a + b 6= 0, consider the...
- 1.2.16: For statements P, Q and R, use truth tables to verify the following...
- 1.2.17: For two statements P and Q, use truth tables to verify the followin...
- 1.2.18: Let P, Q and R be statements. Determine whether the following is tr...
- 1.2.19: Let P, Q and R be statements. Determine whether the following is tr...
- 1.2.20: Let P, Q and R be statements. Determine whether the following is tr...
- 1.2.21: For a real number a, consider the two sentences P: a Q: a > (a) Sta...
- 1.2.22: For integers x and y, let Q(x, y) : (x + 2)2 + 2y2 = (a) Complete t...
Solutions for Chapter 1.2: Negation, Conjunction and Disjunction
Full solutions for Discrete Mathematics | 1st Edition
Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!
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.
Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
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 Fn-1c can be computed with ne/2 multiplications. Revolutionary.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.
Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
A directed graph that has constants Cl, ... , Cm associated with the edges.
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.
Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
Solvable system Ax = b.
The right side b is in the column space of A.
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.