 2.1: Write each of the following sets by listing its elements.(a) A = {x...
 2.2: For a certain set A, the power set of A is P(A) = {, {0},B}, where ...
 2.3: Let A and B be sets. Show that if A 6= B, then P(A) 6= P(B).
 2.4: For S = {3,2, . . . , 2}, list the elements of the set A = {n S : 0...
 2.5: For S = {1, 2, 3}, give an example of two sets A and B such that B...
 2.6: Give an example of three sets A,B and C such that A B, A / C and B ...
 2.7: Determine the power set P(A) for A = {1, 3, 5}.
 2.8: Let A be the set of students in a discrete mathematics course in yo...
 2.9: Let C be the set of professors in a mathematics department who taug...
 2.10: For A = {1, 2}, B = {1, 0, 1} and the universal set U = {2,1, 0, 1,...
 2.11: For a set S, let P_(S) be the set of all proper nonempty subsets of...
 2.12: For S = {{1}, {1, 2}, {1, 2, 3}}, find the power set P(S).1
 2.13: For the set S = {1, 2, . . . , 6}, give an example of a partition P...
 2.14: Determine whether each of the following statements is true or false...
 2.15: Give examples of two sets A and B such that(a) A B and A B.(b) A B ...
 2.16: Give examples of three sets A, B and C such that(a) A B 6 C.(b) A B...
 2.17: For the universal set U = {1, 2, . . . , 10}, draw a Venn diagram f...
 2.18: Let S = {1, 2, . . . , 10}. Give an example of(a) a set A of cardin...
 2.19: Let S = {a, b, c, d, e, f, g}. Give an example of(a) a set A such t...
 2.20: Let Ai = {1, 2, . . . , i} for 1 i 1 Determine the following.(a)10\...
 2.21: For the universal set U = {1, 2, . . . , 8}, let A = {2, 3, 7}, B =...
 2.22: Let S = {1, 3, 5, . . . , 15}. For each of the following sets, desc...
 2.23: Let A = {1, {1}} and B = {1, 2, {2}}. Determine the following.(a) A...
 2.24: For A = {1, 2, . . . , 10}, let P be a collection of k nonempty sub...
 2.25: Determine whether the following statements are true or false.(a) {x...
 2.26: (a) For A = {1, 2, 3}, give an example of three subsets A1,A2 and A...
 2.27: (a) Let A = {1, 2, 3}. Give an example of three distinct subsets A1...
Solutions for Chapter 2: Sets
Full solutions for Discrete Mathematics  1st Edition
ISBN: 9781577667308
Solutions for Chapter 2: Sets
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Discrete Mathematics was written by and is associated to the ISBN: 9781577667308. This textbook survival guide was created for the textbook: Discrete Mathematics, edition: 1. Since 27 problems in chapter 2: Sets have been answered, more than 13010 students have viewed full stepbystep solutions from this chapter. Chapter 2: Sets includes 27 full stepbystep solutions.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

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

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

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].

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.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

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

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

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

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

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

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

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

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

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