 4.1.1: Let S = {2, 5, 17, 27}. Which of the following expressions are true...
 4.1.2: Let B = {x 0 x [ and 1 < x < 2}. Which of the following expressions...
 4.1.3: How many different sets are described here? What are they? {2, 3, 4...
 4.1.4: How many different sets are described here? What are they? {x 0 x =...
 4.1.5: Describe each of the following sets by listing its elements: a. {x ...
 4.1.6: Describe each of the following sets by listing its elements: a. {x ...
 4.1.7: Describe each of the following sets by giving a characterizing prop...
 4.1.8: Describe each of the following sets: a {x 0 x [ and (Eq)(q [ {2, 3}...
 4.1.9: Given the description of a set A as A = {2, 4, 8 }, do you think 16...
 4.1.10: What is the cardinality of each of the following sets? a. S = {a, {...
 4.1.11: Let A = {2, 5, 7} B = {1, 2, 4, 7, 8} C = {7, 8} Which of the follo...
 4.1.12: Let A = {x 0 x [ and 1 < x < 50} B = {x 0 x [ and 1 < x < 50} C = {...
 4.1.13: Let R = {1, 3, , 4.1, 9, 10} T = {1, 3, } S = {{1}, 3, 9, 10} U = {...
 4.1.14: Let R = {1, 3, , 4.1, 9, 10} T = {1, 3, } S = {{1}, 3, 9, 10} U = {...
 4.1.15: Let A = {a, {a}, {{a}}} B = {a} C = {[, {a, {a}}} Which of the foll...
 4.1.16: Let A = {[, {[, {[}}} B = [ C = {[} D = {[, {[}} Which of the follo...
 4.1.17: Let A = {(x, y) 0 (x, y) lies within 3 units of the point (1, 4)} a...
 4.1.18: Let A = {x 0 x [ and x2 4x + 3 < 0} and B = {x 0 x [ and 0 < x < 6}...
 4.1.19: Program QUAD finds and prints solutions to quadratic equations of t...
 4.1.20: Let A = {x 0 cos(x/2) = 0} and B ={x 0 sin x = 0}. Prove that A # B.
 4.1.21: Which of the following statements are true for all sets A, B, and C...
 4.1.22: Which of the following statements are true for all sets A, B, and C...
 4.1.23: Prove that if A # B and B # C, then A # C
 4.1.24: Prove that if A # B, then B # A.
 4.1.25: Prove that for any integer n 2, a set with n elements has n(n 1)/2 ...
 4.1.26: Prove that for any integer n 3, a set with n elements has n(n 1)(n ...
 4.1.27: Find (S) for S = {a}.
 4.1.28: Find (S) for S = {a, b}.
 4.1.29: Find (S) for S = {1, 2, 3, 4}. How many elements do you expect this...
 4.1.30: Find (S) for S = {[}.
 4.1.31: Find (S) for S = {[, {[}, {[, {[}}}
 4.1.32: Find ((S)) for S = {a, b}.
 4.1.33: What can be said about A if (A) = {[, {x}, {y}, {x, y}}?
 4.1.34: What can be said about A if (A) = {[, {a}, {{a}}}?
 4.1.35: Prove that if (A) = (B), then A = B.
 4.1.36: Prove that if A # B, then (A) # (B).
 4.1.37: Solve for x and y. a. (y, x + 2) = (5, 3) b. (2x, y) = (16, 7) c. (...
 4.1.38: a. Recall that ordered pairs must have the property that (x, y) = (...
 4.1.39: Which of the following candidates are binary or unary operations on...
 4.1.40: Which of the following candidates are binary or unary operations on...
 4.1.41: Which of the following candidates are binary or unary operations on...
 4.1.42: Which of the following candidates are binary or unary operations on...
 4.1.43: How many different unary operations can be defined on a set with n ...
 4.1.44: How many different binary operations can be defined on a set with n...
 4.1.45: We have written binary operations in infix notation, where the oper...
 4.1.46: Evaluate the following postfix expressions (see Exercise 45): a 2 4...
 4.1.47: Let A = { p, q, r, s} B = {r, t, v} C = { p, s, t, u} be subsets of...
 4.1.48: Let A = { p, q, r, s} B = {r, t, v} C = { p, s, t, u} be subsets of...
 4.1.49: Let A = {2, 4, 5, 6, 8} B = {1, 4, 5, 9} C = {x 0 x [ and 2 x < 5} ...
 4.1.50: Let A = {2, 4, 5, 6, 8} B = {1, 4, 5, 9} C = {x 0 x [ and 2 x < 5} ...
 4.1.51: Let A = {a, {a}, {{a}}} B = {[, {a}, {a, {a}}} C = {a} be subsets o...
 4.1.52: Let A = {x 0 x is the name of a former president of the United Stat...
 4.1.53: Let S = A B where A = {2, 3, 4} and B = {3, 5}. Which of the follow...
 4.1.54: Let A = {x 0 x is a word that appears before dog in an English lang...
 4.1.55: Consider the following subsets of : A = {x 0 (E y)( y [ and y 4 and...
 4.1.56: Let A = {x 0 x [ and 1 < x 3} B = {x 0 x [ and 2 x 5} Using set ope...
 4.1.57: Consider the following subsets of the set of all students: A = set ...
 4.1.58: Consider the following subsets of the set of all students: A = set ...
 4.1.59: Write the set expression for the desired results of the Web search ...
 4.1.60: Write the set expression for the desired results of the Web search ...
 4.1.61: Write the set expression for the desired results of the Web search ...
 4.1.62: Write the set expression for the desired results of the Web search ...
 4.1.63: Which of the following statements are true for all sets A, B, and C...
 4.1.64: Which of the following statements are true for all sets A, B, and C...
 4.1.65: Which of the following statements are true for all sets A, B, and C...
 4.1.66: Which of the following statements are true for all sets A, B, and C...
 4.1.67: For each of the following statements, find general conditions on se...
 4.1.68: For any finite set S, 0 S 0 denotes the number of elements in S. If...
 4.1.69: Prove that (A d B) # A where A and B are arbitrary sets.
 4.1.70: Prove that A # (A c B) where A and B are arbitrary sets.
 4.1.71: Prove that (A) d (B) = (A d B) where A and B are arbitrary sets.
 4.1.72: Prove that (A) c (B) # (A c B) where A and B are arbitrary sets
 4.1.73: Prove that if A c B = A B, then B = [. (Hint: Do a proof by contrad...
 4.1.74: Prove that if (A B) c (B A) = A c B, then A d B = [. (Hint: Do a pr...
 4.1.75: Prove that if C # B A, then A d C = [.
 4.1.76: Prove that if (A B) c B = A, then B # A.
 4.1.77: Prove that A # B if and only if A d B = [.
 4.1.78: Prove that (A d B) c C = A d (B c C ) if and only if C # A
 4.1.79: a. Draw a Venn diagram to illustrate A ! B. b. For A = {3, 5, 7, 9}...
 4.1.80: a. For an arbitrary set A, what is A ! A? What is [ ! A? b. Prove t...
 4.1.81: Verify the basic set identities on page 234 by showing set inclusio...
 4.1.82: A and B are subsets of a set S. Prove the following set identities ...
 4.1.83: A, B, and C are subsets of a set S. Prove the following set identit...
 4.1.84: A is a subset of a set S. Prove the following set identities: a. A ...
 4.1.85: A, B, and C are subsets of a set S. Prove the following set identit...
 4.1.86: A, B, and C are subsets of a set S. Prove the following set identit...
 4.1.87: The operation of set union can be defined as an nary operation for...
 4.1.88: Using the recursive definition of set union from Exercise 87(b), pr...
 4.1.89: The operation of set intersection can be defined as an nary operat...
 4.1.90: Using the recursive definition of set intersection from Exercise 89...
 4.1.91: Prove that for subsets A1, A2, , An and B of a set S, the following...
 4.1.92: Prove that for subsets A1, A2, , An of a set S, the following gener...
 4.1.93: The operations of set union and set intersection can be extended to...
 4.1.94: According to our use of the word set, if C is a subset of the unive...
 4.1.95: The principle of wellordering says that every nonempty set of posi...
 4.1.96: Prove that the principle of wellordering (see Exercise 95) implies...
 4.1.97: Prove that the set of odd positive integers is denumerable.
 4.1.98: Prove that the set of all integers is denumerable.
 4.1.99: Prove that the set of all finitelength strings of the letter a is ...
 4.1.100: Prove that the set of all finitelength binary strings is denumerable.
 4.1.101: Prove that the set is denumerable.
 4.1.102: In Example 23, the claim was made that 0.249999999 is the same numb...
 4.1.103: Use Cantors diagonalization method to show that the set of all infi...
 4.1.104: Use Cantors diagonalization method to show that the set of all infi...
 4.1.105: Explain why the union of any two denumerable sets is denumerable.
 4.1.106: Explain why any subset of a countable set is countable
 4.1.107: Sets can have sets as elements (see Exercise 13, for example). Let ...
Solutions for Chapter 4.1: Sets
Full solutions for Mathematical Structures for Computer Science  7th Edition
ISBN: 9781429215107
Solutions for Chapter 4.1: Sets
Get Full SolutionsSince 107 problems in chapter 4.1: Sets have been answered, more than 4209 students have viewed full stepbystep solutions from this chapter. Chapter 4.1: Sets includes 107 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Mathematical Structures for Computer Science, edition: 7. Mathematical Structures for Computer Science was written by Patricia and is associated to the ISBN: 9781429215107.

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.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib IIĀ· Condition numbers measure the sensitivity of the output to change in the input.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

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).

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.

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

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

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

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.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

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

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

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

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

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

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.
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