 1.2.1E: Which of the following sets are equal?A = {a, b, c, d}_____________...
 1.2.2E: Write in words how to read each of the following out loud.a. {x ? R...
 1.2.3E: a. Is 4 = {4}?b. How many elements are in the set {3, 4, 3, 5}?c. H...
 1.2.4E: a. Is 2 ? {2}?b. How many elements are in the set {2, 2, 2, 2}?c. H...
 1.2.5E: Which of the following sets are equal?A = {0, 1, 2}B = {x ? R  –1 ...
 1.2.6E: For each integer n, let Tn = {n, n2}. How many elements are in each...
 1.2.7E: Use the setroster notation to indicate the elements in each of the...
 1.2.8E: Let A = {c, d, f, g}, B = {f, j}, and C = {d, g}. Answer each of th...
 1.2.9E: a. Is 3 ? {1, 2, 3}?b. Is 1 ? {1}?c. Is {2} ? {1, 2}?d. Is {3} ? {1...
 1.2.10E: a. Is ((–2)2, –22) = (–22, (–2)2)?b. Is (5, –5) = (–5, 5)?c. Is (8 ...
 1.2.11E: Let A = {w, x, y, z} and B = {a, b}. Use the setroster notation to...
 1.2.12E: Let S = {2, 4, 6} and T = {1,3, 5}. Use the setroster notation to ...
Solutions for Chapter 1.2: Discrete Mathematics with Applications 4th Edition
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 1.2
Get Full SolutionsDiscrete Mathematics with Applications was written by Sieva Kozinsky and is associated to the ISBN: 9780495391326. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4th. Since 12 problems in chapter 1.2 have been answered, more than 24663 students have viewed full stepbystep solutions from this chapter. Chapter 1.2 includes 12 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

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

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

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

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).

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

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.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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.

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

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

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

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

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