 6.2.1E: a. To say that an element is in A ? (B ? C) means that it is in (1)...
 6.2.2E: The following are two proofs that for all sets A and B, A – B ? A. ...
 6.2.3E: The following is a proof that for all sets A, B, and C, if A ? B an...
 6.2.4E: The following is a proof that for all sets A and B, if A ? B, then ...
 6.2.5E: Prove that for all sets A and B, (B – A) = B ? Ac.
 6.2.6E: The following is a proof that for any sets A, B, and C, A ? (B ? C)...
 6.2.7E: Use an element argument to prove each statement. Assume that all se...
 6.2.8E: Use an element argument to prove each statement. Assume that all se...
 6.2.9E: Use an element argument to prove each statement. Assume that all se...
 6.2.10E: Use an element argument to prove each statement. Assume that all se...
 6.2.11E: Use an element argument to prove each statement. Assume that all se...
 6.2.12E: Use an element argument to prove each statement. Assume that all se...
 6.2.13E: Use an element argument to prove each statement. Assume that all se...
 6.2.14E: Use an element argument to prove each statement. Assume that all se...
 6.2.15E: Use an element argument to prove each statement. Assume that all se...
 6.2.16E: Use an element argument to prove each statement. Assume that all se...
 6.2.17E: Use an element argument to prove each statement. Assume that all se...
 6.2.18E: Use an element argument to prove each statement. Assume that all se...
 6.2.19E: Use an element argument to prove each statement. Assume that all se...
 6.2.20E: Find the mistake in the following “proof” that for all sets A, B, a...
 6.2.21E: Find the mistake in the following “proof.”“Theorem:” For all sets A...
 6.2.22E: Find the mistake in the following “proof” that for all sets A and B...
 6.2.23E: Consider the Venn diagram below. a. Illustrate one of the distribut...
 6.2.24E: Fill in the blanks in the following proof that for all sets A and B...
 6.2.25E: Use the element method for proving a set equals the empty set to pr...
 6.2.26E: Use the element method for proving a set equals the empty set to pr...
 6.2.27E: Use the element method for proving a set equals the empty set to pr...
 6.2.28E: Use the element method for proving a set equals the empty set to pr...
 6.2.29E: Use the element method for proving a set equals the empty set to pr...
 6.2.30E: Use the element method for proving a set equals the empty set to pr...
 6.2.31E: Use the element method for proving a set equals the empty set to pr...
 6.2.32E: Use the element method for proving a set equals the empty set to pr...
 6.2.33E: Use the element method for proving a set equals the empty set to pr...
 6.2.34E: Use the element method for proving a set equals the empty set to pr...
 6.2.35E: Use the element method for proving a set equals the empty set to pr...
 6.2.36E: Prove each statement.ExerciseFor all sets A and B,a. (A – B) ? (B –...
 6.2.37E: Prove each statement.ExerciseFor all integers n ? 1, if A and B1, B...
 6.2.38E: Prove each statement.ExerciseFor all integers n ? 1, if A1, A2, A3,...
 6.2.39E: Prove each statement.ExerciseFor all integers n ? 1, if A1, A2, A3,...
 6.2.40E: Prove each statement.ExerciseFor all integers n ? 1, if A and B1, B...
 6.2.41E: Prove each statement.ExerciseFor all integers n ? 1, if A and B1, B...
Solutions for Chapter 6.2: Discrete Mathematics with Applications 4th Edition
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 6.2
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 41 problems in chapter 6.2 have been answered, more than 28126 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4th. Chapter 6.2 includes 41 full stepbystep solutions. Discrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Column space C (A) =
space of all combinations of the columns of A.

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

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

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.

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.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

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.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

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

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

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

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