 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 26882 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: 4th. Discrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326.

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

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

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

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

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

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

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

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

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

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

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.

Kirchhoff's Laws.
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.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

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

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

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

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

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.