 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: 7th. 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 92471 students have viewed full stepbystep solutions from this chapter. Chapter 2.1 includes 47 full stepbystep solutions.

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

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.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

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.

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.

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.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

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).

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

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

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

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.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).