 2.23: In Exercises 114, state whether each statement is true or false. If...
 2.24: In Exercises 114, state whether each statement is true or false. If...
 2.25: In Exercises 114, state whether each statement is true or false. If...
 2.26: In Exercises 114, state whether each statement is true or false. If...
 2.27: In Exercises 114, state whether each statement is true or false. If...
 2.28: In Exercises 114, state whether each statement is true or false. If...
 2.29: In Exercises 114, state whether each statement is true or false. If...
 2.30: In Exercises 114, state whether each statement is true or false. If...
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 2.33: In Exercises 114, state whether each statement is true or false. If...
 2.34: In Exercises 114, state whether each statement is true or false. If...
 2.35: In Exercises 114, state whether each statement is true or false. If...
 2.36: In Exercises 114, state whether each statement is true or false. If...
 2.37: In Exercises 1518, express each set in roster form.Set A is the set...
 2.38: In Exercises 1518, express each set in roster form.Set B is the set...
 2.39: In Exercises 1518, express each set in roster form.C = 5x x N and x...
 2.40: In Exercises 1518, express each set in roster form.D = 5x x N and 8...
 2.41: In Exercises 1922, express each set in setbuilder notation.Set A i...
 2.42: In Exercises 1922, express each set in setbuilder notation.Set B i...
 2.43: In Exercises 1922, express each set in setbuilder notation. Set C ...
 2.44: In Exercises 1922, express each set in setbuilder notation.Set D i...
 2.45: In Exercises 2326, express each set with a written description.A = ...
 2.46: In Exercises 2326, express each set with a written description.B = ...
 2.47: In Exercises 2326, express each set with a written description.C = ...
 2.48: In Exercises 2326, express each set with a written description.D = ...
 2.49: In Exercises 2736, let U = 51, 2, 3, 4, p , 106 A = 51, 3, 5, 76 B ...
 2.50: In Exercises 2736, let U = 51, 2, 3, 4, p , 106 A = 51, 3, 5, 76 B ...
 2.51: In Exercises 2736, let U = 51, 2, 3, 4, p , 106 A = 51, 3, 5, 76 B ...
 2.52: In Exercises 2736, let U = 51, 2, 3, 4, p , 106 A = 51, 3, 5, 76 B ...
 2.53: In Exercises 2736, let U = 51, 2, 3, 4, p , 106 A = 51, 3, 5, 76 B ...
 2.54: In Exercises 2736, let U = 51, 2, 3, 4, p , 106 A = 51, 3, 5, 76 B ...
 2.55: In Exercises 2736, let U = 51, 2, 3, 4, p , 106 A = 51, 3, 5, 76 B ...
 2.56: In Exercises 2736, let U = 51, 2, 3, 4, p , 106 A = 51, 3, 5, 76 B ...
 2.57: In Exercises 2736, let U = 51, 2, 3, 4, p , 106 A = 51, 3, 5, 76 B ...
 2.58: In Exercises 2736, let U = 51, 2, 3, 4, p , 106 A = 51, 3, 5, 76 B ...
 2.59: For the following sets, construct a Venn diagram and place the elem...
 2.60: In Exercises 3843, use Fig. 2.29 to determine the sets. Figure 2.29...
 2.61: In Exercises 3843, use Fig. 2.29 to determine the sets. Figure 2.30...
 2.62: In Exercises 3843, use Fig. 2.29 to determine the sets. Figure 2.31...
 2.63: In Exercises 3843, use Fig. 2.29 to determine the sets. Figure 2.32...
 2.64: In Exercises 3843, use Fig. 2.29 to determine the sets. Figure 2.33...
 2.65: In Exercises 3843, use Fig. 2.29 to determine the sets. Figure 2.34...
 2.66: Construct a Venn diagram to determine whether the following stateme...
 2.67: Construct a Venn diagram to determine whether the following stateme...
 2.68: In Exercises 4652, use the following table, which shows the amount ...
 2.69: In Exercises 4652, use the following table, which shows the amount ...
 2.70: In Exercises 4652, use the following table, which shows the amount ...
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 2.72: In Exercises 4652, use the following table, which shows the amount ...
 2.73: In Exercises 4652, use the following table, which shows the amount ...
 2.74: In Exercises 4652, use the following table, which shows the amount ...
 2.75: Pizza Survey A pizza chain was willing to pay $1 to each person int...
 2.76: Shopping Preferences Visitors to a shopping mall in Atlanta, Georgi...
 2.77: TV Choices TV Guide surveyed 510 subscribers asking which of the fo...
 2.78: In Exercises 56 and 57, show that the sets are infinite by placing ...
 2.79: In Exercises 56 and 57, show that the sets are infinite by placing ...
 2.80: In Exercises 58 and 59, show that each set has cardinal number 0 by...
 2.81: In Exercises 58 and 59, show that each set has cardinal number 0 by...
 2.82: In Exercises 18, state whether each is true or false. If the statem...
 2.83: In Exercises 18, state whether each is true or false. If the statem...
 2.84: In Exercises 18, state whether each is true or false. If the statem...
 2.85: In Exercises 18, state whether each is true or false. If the statem...
 2.86: In Exercises 18, state whether each is true or false. If the statem...
 2.87: In Exercises 18, state whether each is true or false. If the statem...
 2.88: In Exercises 18, state whether each is true or false. If the statem...
 2.89: In Exercises 18, state whether each is true or false. If the statem...
 2.90: In Exercises 9 and 10, use set A = 5x x N and x 6 106. Write set A ...
 2.91: In Exercises 9 and 10, use set A = 5x x N and x 6 106. Write a desc...
 2.92: In Exercises 1116, use the following information. U = 53, 5, 7, 9, ...
 2.93: In Exercises 1116, use the following information. U = 53, 5, 7, 9, ...
 2.94: In Exercises 1116, use the following information. U = 53, 5, 7, 9, ...
 2.95: In Exercises 1116, use the following information. U = 53, 5, 7, 9, ...
 2.96: In Exercises 1116, use the following information. U = 53, 5, 7, 9, ...
 2.97: In Exercises 1116, use the following information. U = 53, 5, 7, 9, ...
 2.98: Using the sets provided for Exercises 1116, draw a Venn diagram ill...
 2.99: Use a Venn diagram to determine whether A y (B x C) = (A y B) x (A ...
 2.100: Water Activities A survey of 155 residents of Lake Placid were aske...
 2.101: Water Activities A survey of 155 residents of Lake Placid were aske...
 2.102: The Wilcox family is considering buying a dog. They have establishe...
 2.103: Read the Mathematics Today feature on page 53. Do research and indi...
 2.104: On Diplomat Row, an area of Washington, DC, there are five houses. ...
Solutions for Chapter 2: Sets
Full solutions for A Survey of Mathematics with Applications  9th Edition
ISBN: 9780321759665
Solutions for Chapter 2: Sets
Get Full SolutionsA Survey of Mathematics with Applications was written by and is associated to the ISBN: 9780321759665. Chapter 2: Sets includes 82 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: A Survey of Mathematics with Applications, edition: 9. Since 82 problems in chapter 2: Sets have been answered, more than 78634 students have viewed full stepbystep solutions from this chapter.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

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.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

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).

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].

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

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

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.

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 inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

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.

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

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

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

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.

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.

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!

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.