 2.2.1E: Let A be the set of students who live within one mile of school and...
 2.2.2E: Suppose that A is the set of sophomores at your school and B is the...
 2.2.3E: Let A = {1, 2, 3, 4, 5) and B = {0, 3, 6}. Finda) A ? B.___________...
 2.2.4E: Let A = {a, b, c, d, e} and B = {a, b, c, d, e, f, g, h} Finda) A ?...
 2.2.5E: Prove the complementation law in Table 1 by showing that
 2.2.6E: Prove the identity laws in Table 1 by showing thata) A ? ? = A.____...
 2.2.7E: Prove the domination laws in Table 1 by showing thata) A ? U = U.__...
 2.2.8E: Prove the idempotent laws in Table 1 by showing thata) A ? A = A.__...
 2.2.9E: Prove the complement laws in Table 1 by showing thata) ____________...
 2.2.10E: Show thata) A — ? = A.________________b) ? – A = ?.
 2.2.11E: Let A and B be sets. Prove the commutative laws from Table 1 by sho...
 2.2.12E: Prove the first absorption law from Table 1 by showing that if A an...
 2.2.13E: Prove the second absorption law from Table 1 by showing that if A a...
 2.2.14E: Find the sets A and B if A – B = {1, 5, 7, 8}, B – A = {2, 10}, and...
 2.2.15E: Prove the second De Morgan law in Table l by showing that if A and ...
 2.2.16E: ?ProBlem 16ELet A And B Be sets. Show thAtA) (A ?B) ?A.____________...
 2.2.17E: Show that if A,b, and C are sets, then a) by showing each side is a...
 2.2.18E: Leta, b, and C be sets. Show thata) (A ? B) ? (A ? B ? C)._________...
 2.2.19E: Show that if A and B are sets, then
 2.2.20E: Show that if A andb are sets with A ?b, thena) A ?b = b.___________...
 2.2.21E: Prove the first associative law from Table 1 by showing that if A.b...
 2.2.22E: Prove the second associative law from Table 1 by showing that if A,...
 2.2.23E: Prove the first distributive law from Table 1 by showing that if A....
 2.2.24E: Let A,b. and C be sets. Show that (A —b) — C = (A – C)(B – C).
 2.2.25E: Let A = {0, 2, 4, 6, 8, 10}. B = {0, 1, 2, 3, 4, 5, 6}. and C = {4,...
 2.2.26E: Draw the Venn diagrams for each of these combinations of the sets A...
 2.2.27E: Draw the Venn diagrams for each of these combinations of the sets A...
 2.2.28E: Draw the Venn diagrams for each of these combinations of the sets A...
 2.2.29E: What can you say about the sets A and B if we know thata) A ? B = A...
 2.2.30E: Can you conclude that A = B if A, B. and C arc sets such thata) A ?...
 2.2.31E: Let A and B be subsets of a universal set U. Show that A ? B if and...
 2.2.32E: Find the symmetric difference of {1, 3, 5} and {1, 2, 3}.
 2.2.33E: Find the symmetric difference of the set of computer science majors...
 2.2.34E: Draw a Venn diagram for the symmetric difference of the setsa andb.
 2.2.35E: Show thata?b = (a ?b) – (A ?b).
 2.2.36E: Show that A ? B = (A – B) ? (B – A).
 2.2.37E: Show that ifa is a subset of a universal setu, thena) A ? A = ?.___...
 2.2.38E: Show that ifa andb are sets, thena) A? B = B? A.________________b) ...
 2.2.39E: What can you say about the setsa andb ifa?b = a?
 2.2.40E: Determine whether the symmetric difference is associative; that is,...
 2.2.41E: The symmetric difference of A And B, denoted by A ? B, is the set c...
 2.2.42E: The symmetric difference of A And B, denoted by A ? B, is the set c...
 2.2.43E: The symmetric difference of A And B, denoted by A ? B, is the set c...
 2.2.44E: Show that if A and B are finite sets, then A ? B is a finite set.
 2.2.45E: Show that if A is an infinite set, then whenever B is a set. A ? B ...
 2.2.46E: Show that ifa, b, and C are sets, then (This is a special case of t...
 2.2.47E: Let Ai, = {1, 2, 3.....,i} for i = 1, 2, 3,....,Finda) ____________...
 2.2.48E: Let Ai = {....,–2,–1, 0, 1,....,i}. Finda) ________________b)
 2.2.49E: Let Ai be the set of all nonempty bit strings (that is, bit strings...
 2.2.50E: Find and if for every positive integer i,a) Ai = {i, i + l, i + 2,...
 2.2.51E: Find and if for every positive integer i,a) Ai = {?i, –i + 1,...,–1...
 2.2.52E: Suppose that the universal set isu = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10...
 2.2.53E: Using the same universal set as in the last problem, find the set s...
 2.2.54E: What subsets of a finite universal set do these bit strings represe...
 2.2.55E: What is the bit string corresponding to the difference of two sets?
 2.2.56E: What is the bit string corresponding to the symmetric difference of...
 2.2.57E: Show how bitwise operations on bit strings can be used to find thes...
 2.2.58E: How can the union and intersection of n sets that all are subsets o...
 2.2.59E: Find the successors of the following sets.a) {1, 2, 3}_____________...
 2.2.60E: How many elements does the successor of a set with n elements have?...
 2.2.61E: Let A and B be the multisets {3 a, 2 b, 1 c) and {2 a. 3 b. 4 d), r...
 2.2.62E: Suppose that A is the multiset that has as its elements the types o...
 2.2.63E: The complement of a fuzzy set S is the set , with the degree of the...
 2.2.64E: Fuzzy Sets are used in artificial intelligence. Each elememt in the...
 2.2.65E: The intersection of two fuzzy sets S and T is the fuzzy set S ? T. ...
Solutions for Chapter 2.2: Discrete Mathematics and Its Applications 7th Edition
Full solutions for Discrete Mathematics and Its Applications  7th Edition
ISBN: 9780073383095
Solutions for Chapter 2.2
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 2.2 includes 65 full stepbystep solutions. Since 65 problems in chapter 2.2 have been answered, more than 252253 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 7. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095.

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.

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).

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

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.

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.

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.

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

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

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

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

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

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

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

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.