 2.1.1: Write each of the following sets by listing its elements. (a) A = {...
 2.1.2: Let S = {3,2,1, 0, 1, 2, 3, 4}. Describe each of the following subs...
 2.1.3: Describe each of the following sets in symbols. (a) The set of all ...
 2.1.4: Describe each of the following sets in words. (a) {x R : x5 3x3 x2 ...
 2.1.5: For each of the following sets, indicate whether 2 is an element of...
 2.1.6: For each of the following sets, indicate whether a is an element of...
 2.1.7: Determine the cardinality of each of the following sets. (a) A = {a...
 2.1.8: Determine the cardinality of each of the following sets. (a) A = {{...
 2.1.9: Which of the following pairs of sets are equal? (a) A1 = {1, 2, 3} ...
 2.1.10: Which of the following sets are equal? (a) A = {n Z : n < 2}. (b)...
 2.1.11: Determine whether each of the following statements is true or false...
 2.1.12: Let A = {x Z : x 2} and C = {x Z : 2 x 4}. Find all sets B such t...
 2.1.13: Draw a Venn diagram for two general sets C and D and shade the regi...
 2.1.14: Draw a Venn diagram for two general sets C and D and shade the regi...
 2.1.15: Give an example of three sets A,B and C such that none of these thr...
 2.1.16: Give an example of three sets A,B and C such that A B, B C, C 6 A a...
 2.1.17: Give an example of three sets A,B and C such that A B, A C and C B ...
 2.1.18: Give an example of two sets A and B such that A B and A B. 1
 2.1.19: Give examples of three sets A,B and C such that (a) A B C. (b) A B,...
 2.1.20: Determine the power set of each of the following sets. (a) . (b) {}...
 2.1.21: Determine the power set of each of the following sets. (a) A = {, a...
 2.1.22: Determine the power set of each of the following sets. (a) A = {0, ...
 2.1.23: How many elements are in P(A) if A = {1, 2, 3, 4, 5}? 2
 2.1.24: How many elements are in P(A) if A = {n Z : n 5}? 2
 2.1.25: Let A = {1, 2, . . . , 10}. Give an example of a set B such that (a...
 2.1.26: For n N, two sets A and C have the property that A C, A = n and ...
Solutions for Chapter 2.1: Sets and Subsets
Full solutions for Discrete Mathematics  1st Edition
ISBN: 9781577667308
Solutions for Chapter 2.1: Sets and Subsets
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 26 problems in chapter 2.1: Sets and Subsets have been answered, more than 13675 students have viewed full stepbystep solutions from this chapter. 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. Chapter 2.1: Sets and Subsets includes 26 full stepbystep solutions.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

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!

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.

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

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

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.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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

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).

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

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

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

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

Stiffness matrix
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.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.