 2.3.1: In 14, describe a universal set U that includes all elements in the...
 2.3.2: In 14, describe a universal set U that includes all elements in the...
 2.3.3: In 14, describe a universal set U that includes all elements in the...
 2.3.4: In 14, describe a universal set U that includes all elements in the...
 2.3.5: In 58, let U = 5a, b, c, d, e, f, g6, A = 5a, b, f, g6, B = 5c, d, ...
 2.3.6: In 58, let U = 5a, b, c, d, e, f, g6, A = 5a, b, f, g6, B = 5c, d, ...
 2.3.7: In 58, let U = 5a, b, c, d, e, f, g6, A = 5a, b, f, g6, B = 5c, d, ...
 2.3.8: In 58, let U = 5a, b, c, d, e, f, g6, A = 5a, b, f, g6, B = 5c, d, ...
 2.3.9: In 912, let U = 51, 2, 3, 4, c, 206, A = 51, 2, 3, 4, 56, B = 56, 7...
 2.3.10: In 912, let U = 51, 2, 3, 4, c, 206, A = 51, 2, 3, 4, 56, B = 56, 7...
 2.3.11: In 912, let U = 51, 2, 3, 4, c, 206, A = 51, 2, 3, 4, 56, B = 56, 7...
 2.3.12: In 912, let U = 51, 2, 3, 4, c, 206, A = 51, 2, 3, 4, 56, B = 56, 7...
 2.3.13: In 1316, let U = 51, 2, 3, 4, c6, A = 51, 2, 3, 4, c, 206, B = 51, ...
 2.3.14: In 1316, let U = 51, 2, 3, 4, c6, A = 51, 2, 3, 4, c, 206, B = 51, ...
 2.3.15: In 1316, let U = 51, 2, 3, 4, c6, A = 51, 2, 3, 4, c, 206, B = 51, ...
 2.3.16: In 1316, let U = 51, 2, 3, 4, c6, A = 51, 2, 3, 4, c, 206, B = 51, ...
 2.3.17: In 1740, let U = 51, 2, 3, 4, 5, 6, 76 A = 51, 3, 5, 76 B = 51, 2, ...
 2.3.18: In 1740, let U = 51, 2, 3, 4, 5, 6, 76 A = 51, 3, 5, 76 B = 51, 2, ...
 2.3.19: In 1740, let U = 51, 2, 3, 4, 5, 6, 76 A = 51, 3, 5, 76 B = 51, 2, ...
 2.3.20: In 1740, let U = 51, 2, 3, 4, 5, 6, 76 A = 51, 3, 5, 76 B = 51, 2, ...
 2.3.21: In 1740, let U = 51, 2, 3, 4, 5, 6, 76 A = 51, 3, 5, 76 B = 51, 2, ...
 2.3.22: In 1740, let U = 51, 2, 3, 4, 5, 6, 76 A = 51, 3, 5, 76 B = 51, 2, ...
 2.3.23: In 1740, let U = 51, 2, 3, 4, 5, 6, 76 A = 51, 3, 5, 76 B = 51, 2, ...
 2.3.24: In 1740, let U = 51, 2, 3, 4, 5, 6, 76 A = 51, 3, 5, 76 B = 51, 2, ...
 2.3.25: In 1740, let U = 51, 2, 3, 4, 5, 6, 76 A = 51, 3, 5, 76 B = 51, 2, ...
 2.3.26: In 1740, let U = 51, 2, 3, 4, 5, 6, 76 A = 51, 3, 5, 76 B = 51, 2, ...
 2.3.27: In 1740, let U = 51, 2, 3, 4, 5, 6, 76 A = 51, 3, 5, 76 B = 51, 2, ...
 2.3.28: In 1740, let U = 51, 2, 3, 4, 5, 6, 76 A = 51, 3, 5, 76 B = 51, 2, ...
 2.3.29: In 1740, let U = 51, 2, 3, 4, 5, 6, 76 A = 51, 3, 5, 76 B = 51, 2, ...
 2.3.30: In 1740, let U = 51, 2, 3, 4, 5, 6, 76 A = 51, 3, 5, 76 B = 51, 2, ...
 2.3.31: In 1740, let U = 51, 2, 3, 4, 5, 6, 76 A = 51, 3, 5, 76 B = 51, 2, ...
 2.3.32: In 1740, let U = 51, 2, 3, 4, 5, 6, 76 A = 51, 3, 5, 76 B = 51, 2, ...
 2.3.33: In 1740, let U = 51, 2, 3, 4, 5, 6, 76 A = 51, 3, 5, 76 B = 51, 2, ...
 2.3.34: In 1740, let U = 51, 2, 3, 4, 5, 6, 76 A = 51, 3, 5, 76 B = 51, 2, ...
 2.3.35: In 1740, let U = 51, 2, 3, 4, 5, 6, 76 A = 51, 3, 5, 76 B = 51, 2, ...
 2.3.36: In 1740, let U = 51, 2, 3, 4, 5, 6, 76 A = 51, 3, 5, 76 B = 51, 2, ...
 2.3.37: In 1740, let U = 51, 2, 3, 4, 5, 6, 76 A = 51, 3, 5, 76 B = 51, 2, ...
 2.3.38: In 1740, let U = 51, 2, 3, 4, 5, 6, 76 A = 51, 3, 5, 76 B = 51, 2, ...
 2.3.39: In 1740, let U = 51, 2, 3, 4, 5, 6, 76 A = 51, 3, 5, 76 B = 51, 2, ...
 2.3.40: In 1740, let U = 51, 2, 3, 4, 5, 6, 76 A = 51, 3, 5, 76 B = 51, 2, ...
 2.3.41: In 4166, let U = 5a, b, c, d, e, f, g, h6 A = 5a, g, h6 B = 5b, g, ...
 2.3.42: In 4166, let U = 5a, b, c, d, e, f, g, h6 A = 5a, g, h6 B = 5b, g, ...
 2.3.43: In 4166, let U = 5a, b, c, d, e, f, g, h6 A = 5a, g, h6 B = 5b, g, ...
 2.3.44: In 4166, let U = 5a, b, c, d, e, f, g, h6 A = 5a, g, h6 B = 5b, g, ...
 2.3.45: In 4166, let U = 5a, b, c, d, e, f, g, h6 A = 5a, g, h6 B = 5b, g, ...
 2.3.46: In 4166, let U = 5a, b, c, d, e, f, g, h6 A = 5a, g, h6 B = 5b, g, ...
 2.3.47: In 4166, let U = 5a, b, c, d, e, f, g, h6 A = 5a, g, h6 B = 5b, g, ...
 2.3.48: In 4166, let U = 5a, b, c, d, e, f, g, h6 A = 5a, g, h6 B = 5b, g, ...
 2.3.49: In 4166, let U = 5a, b, c, d, e, f, g, h6 A = 5a, g, h6 B = 5b, g, ...
 2.3.50: In 4166, let U = 5a, b, c, d, e, f, g, h6 A = 5a, g, h6 B = 5b, g, ...
 2.3.51: In 4166, let U = 5a, b, c, d, e, f, g, h6 A = 5a, g, h6 B = 5b, g, ...
 2.3.52: In 4166, let U = 5a, b, c, d, e, f, g, h6 A = 5a, g, h6 B = 5b, g, ...
 2.3.53: In 4166, let U = 5a, b, c, d, e, f, g, h6 A = 5a, g, h6 B = 5b, g, ...
 2.3.54: In 4166, let U = 5a, b, c, d, e, f, g, h6 A = 5a, g, h6 B = 5b, g, ...
 2.3.55: In 4166, let U = 5a, b, c, d, e, f, g, h6 A = 5a, g, h6 B = 5b, g, ...
 2.3.56: In 4166, let U = 5a, b, c, d, e, f, g, h6 A = 5a, g, h6 B = 5b, g, ...
 2.3.57: In 4166, let U = 5a, b, c, d, e, f, g, h6 A = 5a, g, h6 B = 5b, g, ...
 2.3.58: In 4166, let U = 5a, b, c, d, e, f, g, h6 A = 5a, g, h6 B = 5b, g, ...
 2.3.59: In 4166, let U = 5a, b, c, d, e, f, g, h6 A = 5a, g, h6 B = 5b, g, ...
 2.3.60: In 4166, let U = 5a, b, c, d, e, f, g, h6 A = 5a, g, h6 B = 5b, g, ...
 2.3.61: In 4166, let U = 5a, b, c, d, e, f, g, h6 A = 5a, g, h6 B = 5b, g, ...
 2.3.62: In 4166, let U = 5a, b, c, d, e, f, g, h6 A = 5a, g, h6 B = 5b, g, ...
 2.3.63: In 4166, let U = 5a, b, c, d, e, f, g, h6 A = 5a, g, h6 B = 5b, g, ...
 2.3.64: In 4166, let U = 5a, b, c, d, e, f, g, h6 A = 5a, g, h6 B = 5b, g, ...
 2.3.65: In 4166, let U = 5a, b, c, d, e, f, g, h6 A = 5a, g, h6 B = 5b, g, ...
 2.3.66: In 4166, let U = 5a, b, c, d, e, f, g, h6 A = 5a, g, h6 B = 5b, g, ...
 2.3.67: In 6778, use the Venn diagram to represent each set in roster form.A
 2.3.68: In 6778, use the Venn diagram to represent each set in roster form.B
 2.3.69: In 6778, use the Venn diagram to represent each set in roster form.U
 2.3.70: In 6778, use the Venn diagram to represent each set in roster form.A B
 2.3.71: In 6778, use the Venn diagram to represent each set in roster form.A B
 2.3.72: In 6778, use the Venn diagram to represent each set in roster form.. A
 2.3.73: In 6778, use the Venn diagram to represent each set in roster form.B
 2.3.74: In 6778, use the Venn diagram to represent each set in roster form....
 2.3.75: In 6778, use the Venn diagram to represent each set in roster form....
 2.3.76: In 6778, use the Venn diagram to represent each set in roster form.A B
 2.3.77: In 6778, use the Venn diagram to represent each set in roster form.A B
 2.3.78: In 6778, use the Venn diagram to represent each set in roster form.A B
 2.3.79: In 7992, use the Venn diagram to determine each set or cardinality....
 2.3.80: In 7992, use the Venn diagram to determine each set or cardinality....
 2.3.81: In 7992, use the Venn diagram to determine each set or cardinality....
 2.3.82: In 7992, use the Venn diagram to determine each set or cardinality....
 2.3.83: In 7992, use the Venn diagram to determine each set or cardinality....
 2.3.84: In 7992, use the Venn diagram to determine each set or cardinality....
 2.3.85: In 7992, use the Venn diagram to determine each set or cardinality....
 2.3.86: In 7992, use the Venn diagram to determine each set or cardinality....
 2.3.87: In 7992, use the Venn diagram to determine each set or cardinality....
 2.3.88: In 7992, use the Venn diagram to determine each set or cardinality....
 2.3.89: In 7992, use the Venn diagram to determine each set or cardinality....
 2.3.90: In 7992, use the Venn diagram to determine each set or cardinality....
 2.3.91: In 7992, use the Venn diagram to determine each set or cardinality....
 2.3.92: In 7992, use the Venn diagram to determine each set or cardinality....
 2.3.93: Use the formula for the cardinal number of the union of two sets to...
 2.3.94: Use the formula for the cardinal number of the union of two sets to...
 2.3.95: Use the formula for the cardinal number of the union of two sets to...
 2.3.96: Use the formula for the cardinal number of the union of two sets to...
 2.3.97: In 97104, let U = 5xxN and x 6 96 A = 5xx is an odd natural number ...
 2.3.98: In 97104, let U = 5xxN and x 6 96 A = 5xx is an odd natural number ...
 2.3.99: In 97104, let U = 5xxN and x 6 96 A = 5xx is an odd natural number ...
 2.3.100: In 97104, let U = 5xxN and x 6 96 A = 5xx is an odd natural number ...
 2.3.101: In 97104, let U = 5xxN and x 6 96 A = 5xx is an odd natural number ...
 2.3.102: In 97104, let U = 5xxN and x 6 96 A = 5xx is an odd natural number ...
 2.3.103: In 97104, let U = 5xxN and x 6 96 A = 5xx is an odd natural number ...
 2.3.104: In 97104, let U = 5xxN and x 6 96 A = 5xx is an odd natural number ...
 2.3.105: In 105108, use the Venn diagram to determine each set or cardinalit...
 2.3.106: In 105108, use the Venn diagram to determine each set or cardinalit...
 2.3.107: In 105108, use the Venn diagram to determine each set or cardinalit...
 2.3.108: In 105108, use the Venn diagram to determine each set or cardinalit...
 2.3.109: A math tutor working with a small group of students asked each stud...
 2.3.110: A math tutor working with a small group of students asked each stud...
 2.3.111: A math tutor working with a small group of students asked each stud...
 2.3.112: A math tutor working with a small group of students asked each stud...
 2.3.113: A math tutor working with a small group of students asked each stud...
 2.3.114: A math tutor working with a small group of students asked each stud...
 2.3.115: A math tutor working with a small group of students asked each stud...
 2.3.116: A math tutor working with a small group of students asked each stud...
 2.3.117: In 117122, use the information in the graph to place the indicated ...
 2.3.118: In 117122, use the information in the graph to place the indicated ...
 2.3.119: In 117122, use the information in the graph to place the indicated ...
 2.3.120: In 117122, use the information in the graph to place the indicated ...
 2.3.121: In 117122, use the information in the graph to place the indicated ...
 2.3.122: In 117122, use the information in the graph to place the indicated ...
 2.3.123: A palindromic number is a natural number whose value does not chang...
 2.3.124: A palindromic number is a natural number whose value does not chang...
 2.3.125: A palindromic number is a natural number whose value does not chang...
 2.3.126: A palindromic number is a natural number whose value does not chang...
 2.3.127: A palindromic number is a natural number whose value does not chang...
 2.3.128: A palindromic number is a natural number whose value does not chang...
 2.3.129: A palindromic number is a natural number whose value does not chang...
 2.3.130: A palindromic number is a natural number whose value does not chang...
 2.3.131: A palindromic number is a natural number whose value does not chang...
 2.3.132: A palindromic number is a natural number whose value does not chang...
 2.3.133: The bar graph shows the percentage of Americans who were not too ha...
 2.3.134: The bar graph shows the percentage of Americans who were not too ha...
 2.3.135: The bar graph shows the percentage of Americans who were not too ha...
 2.3.136: The bar graph shows the percentage of Americans who were not too ha...
 2.3.137: The bar graph shows the percentage of Americans who were not too ha...
 2.3.138: The bar graph shows the percentage of Americans who were not too ha...
 2.3.139: A winter resort took a poll of its 350 visitors to see which winter...
 2.3.140: A pet store surveyed 200 pet owners and obtained the following resu...
 2.3.141: Describe what is meant by a universal set. Provide an example.
 2.3.142: What is a Venn diagram and how is it used?
 2.3.143: Describe the Venn diagram for two disjoint sets. How does this diag...
 2.3.144: Describe the Venn diagram for proper subsets. How does this diagram...
 2.3.145: Describe the Venn diagram for two equal sets. How does this diagram...
 2.3.146: Describe the Venn diagram for two sets with common elements. How do...
 2.3.147: Describe what is meant by the complement of a set.
 2.3.148: Is it possible to find a sets complement if a universal set is not ...
 2.3.149: Describe what is meant by the intersection of two sets. Give an exa...
 2.3.150: Describe what is meant by the union of two sets. Give an example.
 2.3.151: Describe how to find the cardinal number of the union of two finite...
 2.3.152: Make Sense? In 152155, determine whether each statement makes sense...
 2.3.153: Make Sense? In 152155, determine whether each statement makes sense...
 2.3.154: Make Sense? In 152155, determine whether each statement makes sense...
 2.3.155: Make Sense? In 152155, determine whether each statement makes sense...
 2.3.156: In 156163, determine whether each statement is true or false. If th...
 2.3.157: In 156163, determine whether each statement is true or false. If th...
 2.3.158: In 156163, determine whether each statement is true or false. If th...
 2.3.159: In 156163, determine whether each statement is true or false. If th...
 2.3.160: In 156163, determine whether each statement is true or false. If th...
 2.3.161: In 156163, determine whether each statement is true or false. If th...
 2.3.162: In 156163, determine whether each statement is true or false. If th...
 2.3.163: In 156163, determine whether each statement is true or false. If th...
 2.3.164: In 164167, assume A B. Draw a Venn diagram that correctly illustrat...
 2.3.165: In 164167, assume A B. Draw a Venn diagram that correctly illustrat...
 2.3.166: In 164167, assume A B. Draw a Venn diagram that correctly illustrat...
 2.3.167: In 164167, assume A B. Draw a Venn diagram that correctly illustrat...
Solutions for Chapter 2.3: Venn Diagrams and Set Operations
Full solutions for Thinking Mathematically  6th Edition
ISBN: 9780321867322
Solutions for Chapter 2.3: Venn Diagrams and Set Operations
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Thinking Mathematically, edition: 6. Chapter 2.3: Venn Diagrams and Set Operations includes 167 full stepbystep solutions. Thinking Mathematically was written by and is associated to the ISBN: 9780321867322. Since 167 problems in chapter 2.3: Venn Diagrams and Set Operations have been answered, more than 65618 students have viewed full stepbystep solutions from this chapter.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

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).

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

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.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

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

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

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

Iterative method.
A sequence of steps intended to approach the desired solution.

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.

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

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

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).

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.

Outer product uv T
= column times row = rank one matrix.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

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

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.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.