 2.1.1E: List the members of these sets.a) {x  x is a real number such that...
 2.1.2E: Use set builder notation to give a description of each of these set...
 2.1.3E: For each of these pairs of sets, determine whether (he first is a s...
 2.1.4E: For each of these pairs of sets, determine whether the first is a s...
 2.1.5E: Determine whether each of these pairs of sets are equal.a) {1, 3, 3...
 2.1.6E: Suppose that A = {2, 4, 6}. B = {2, 6}. C = {4, 6}. and D = {4, 6, ...
 2.1.7E: For each of the following sets, determine whether 2 is an element o...
 2.1.8E: For each of the sets in Exercise 7. determine whether {2} is an ele...
 2.1.9E: Determine whether each of these statements is true or false.a) 0 ? ...
 2.1.10E: Determine whether these statements are true or falsea) ? ? (?)_____...
 2.1.11E: Determine whether each of these statements is true or false.a) x ? ...
 2.1.12E: Use a Venn diagram to illustrate the subset of odd integers in the ...
 2.1.13E: Use a Venn diagram lo illustrate the set of all months of the year ...
 2.1.14E: Use a Venn diagram to illustrate the relationship A ?. B and B ? C.
 2.1.15E: Use a Venn diagram to illustrate the relationships A ? B and B ? C.
 2.1.16E: Use a Venn diagram to illustrate the relationships A ? B and A ? C.
 2.1.17E: Suppose that A, B, and C are sets such that A ? B and B ? C. Show t...
 2.1.18E: Find two sets A and B such that A ? B and A ? B.
 2.1.19E: What is the cardinality of each of these sets?a) {a}_______________...
 2.1.20E: What is the cardinality of each of these sets?a) ?________________b...
 2.1.21E: Find the power set of each of these sets, where a and b are distinc...
 2.1.22E: Can you conclude that A = B if A and B are two sets with the same p...
 2.1.23E: How many elements does each of these sets have where a and b are di...
 2.1.24E: Determine whether each of these sets is the power set of a set. whe...
 2.1.25E: Prove that P(A) ? P(B) if and only if A ? B.
 2.1.26E: Show that if A ? C and B ? D. then ,A × B ? C × D
 2.1.27E: Let .A = {a. b. c. d} and B = {y, z). Finda) .A × B._______________...
 2.1.28E: What is the Cartesian product A × B. where A is the set of courses ...
 2.1.29E: What is the Cartesian product A × B × C. where A is the set of all ...
 2.1.30E: Suppose that A × B = where, A and B are sets. What can you conclude?
 2.1.31E: Let A be a set. Show that ? × A = A × ? = ?
 2.1.32E: Let .A = {a, b, c}, B = {x, y). and C = {0. 1}. Finda) A × B × C.__...
 2.1.33E: Find, A2 ifa) A = {0, 1, 3}.________________b) A = {1, 2, a, b}.
 2.1.34E: Find A3 ifa) A ={a}________________b) A = (0, a).
 2.1.35E: How many different elements does A × B have if A has m elements and...
 2.1.36E: How many different elements does A × B × C have if A has m elements...
 2.1.37E: How many different elements does A" have when A has m elements and ...
 2.1.38E: Show that A × B ? B × A. when A and B are nonempty, unless A = B.
 2.1.39E: Explain why A × B × C and (A × B) × C are not the same.
 2.1.40E: Explain why (A × B) x (C × D) and A × (B × C) × D are not the same.
 2.1.41E: Translate each of these quantifications into English and determine ...
 2.1.42E: Translate each of these quantifications into English and determine ...
 2.1.43E: Find the truth set of each of these predicates where the domain is ...
 2.1.44E: Find the truth set of each of these predicates where the domain is ...
 2.1.45E: The defining property of an ordered pair is that two ordered pairs ...
 2.1.46E: This exercise presents Russell’s paradox. Let S be the set that con...
 2.1.47E: Describe a procedure for listing all the subsets of a finite set.
Solutions for Chapter 2.1: Discrete Mathematics and Its Applications 7th Edition
Full solutions for Discrete Mathematics and Its Applications  7th Edition
ISBN: 9780073383095
Solutions for Chapter 2.1
Get Full SolutionsThis textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 7. This expansive textbook survival guide covers the following chapters and their solutions. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095. Since 47 problems in chapter 2.1 have been answered, more than 163977 students have viewed full stepbystep solutions from this chapter. Chapter 2.1 includes 47 full stepbystep solutions.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

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

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].

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).

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

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

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.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

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

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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.

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

Solvable system Ax = b.
The right side b is in the column space of A.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

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.

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