 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 113380 students have viewed full stepbystep solutions from this chapter. Chapter 2.1 includes 47 full stepbystep solutions.

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!

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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.

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

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

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.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

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

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

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.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

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

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

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

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

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