 1.2.1: Which of the following sets are equal? A = {a, b, c, d} B = {d, e, ...
 1.2.2: Write in words how to read each of the following out loud. a. {x R+...
 1.2.3: a. Is 4 = {4}? b. How many elements are in the set {3, 4, 3, 5}? c....
 1.2.4: a. Is 2 {2}? b. How many elements are in the set {2, 2, 2, 2}? c. H...
 1.2.5: Which of the following sets are equal? A = {0, 1, 2} B = {x R  1 x...
 1.2.6: For each integer n, let Tn = {n, n2}. How many elements are in each...
 1.2.7: For each integer n, let Tn = {n, n2}. How many elements are in each...
 1.2.8: Let A = {c, d, f, g}, B = { f, j}, and C = {d, g}. Answer each of t...
 1.2.9: a. Is 3 {1, 2, 3}? b. Is 1 {1}? c. Is {2}{1, 2}? d. Is {3}{1,{2},{3...
 1.2.10: a. Is ((2)2, 22) = (22, (2)2)? b. Is (5, 5) = (5, 5)? c. Is 8 9, 3 ...
 1.2.11: Let A = {w, x, y,z} and B = {a, b}. Use the setroster notation to ...
 1.2.12: Let S = {2, 4, 6} and T = {1, 3, 5}. Use the setroster notation to...
Solutions for Chapter 1.2: The Language of Sets
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 1.2: The Language of Sets
Get Full SolutionsChapter 1.2: The Language of Sets includes 12 full stepbystep solutions. Since 12 problems in chapter 1.2: The Language of Sets have been answered, more than 56912 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. 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.

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

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

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. 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.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

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

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

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

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

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

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

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)jI and det V = product of (Xk  Xi) for k > i.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.