 6.3.1: For each of 14 find a counterexample to show that the statement is ...
 6.3.2: For each of 14 find a counterexample to show that the statement is ...
 6.3.3: For each of 14 find a counterexample to show that the statement is ...
 6.3.4: For each of 14 find a counterexample to show that the statement is ...
 6.3.5: For each of 521 prove each statement that is true and find a counte...
 6.3.6: For each of 521 prove each statement that is true and find a counte...
 6.3.7: For each of 521 prove each statement that is true and find a counte...
 6.3.8: For each of 521 prove each statement that is true and find a counte...
 6.3.9: For each of 521 prove each statement that is true and find a counte...
 6.3.10: For each of 521 prove each statement that is true and find a counte...
 6.3.11: For each of 521 prove each statement that is true and find a counte...
 6.3.12: For each of 521 prove each statement that is true and find a counte...
 6.3.13: For each of 521 prove each statement that is true and find a counte...
 6.3.14: For each of 521 prove each statement that is true and find a counte...
 6.3.15: For each of 521 prove each statement that is true and find a counte...
 6.3.16: For each of 521 prove each statement that is true and find a counte...
 6.3.17: For each of 521 prove each statement that is true and find a counte...
 6.3.18: For each of 521 prove each statement that is true and find a counte...
 6.3.19: For each of 521 prove each statement that is true and find a counte...
 6.3.20: For each of 521 prove each statement that is true and find a counte...
 6.3.21: For each of 521 prove each statement that is true and find a counte...
 6.3.22: Write a negation for each of the following statements. Indicate whi...
 6.3.23: Let S = {a, b, c} and for each integer i = 0, 1, 2, 3, let Si be th...
 6.3.24: Let S = {a, b, c} and let Sa be the set of all subsets of S that co...
 6.3.25: Let A = {t, u, v, w} and let S1 be the set of all subsets of A that...
 6.3.26: The following problem, devised by Ginger Bolton, appeared in the Ja...
 6.3.27: In 27 and 28 supply a reason for each step in the derivation.
 6.3.28: In 27 and 28 supply a reason for each step in the derivation.
 6.3.29: Some steps are missing from the following proof that for all sets (...
 6.3.30: In 3040, construct an algebraic proof for the given statement. Cite...
 6.3.31: In 3040, construct an algebraic proof for the given statement. Cite...
 6.3.32: In 3040, construct an algebraic proof for the given statement. Cite...
 6.3.33: In 3040, construct an algebraic proof for the given statement. Cite...
 6.3.34: In 3040, construct an algebraic proof for the given statement. Cite...
 6.3.35: In 3040, construct an algebraic proof for the given statement. Cite...
 6.3.36: In 3040, construct an algebraic proof for the given statement. Cite...
 6.3.37: In 3040, construct an algebraic proof for the given statement. Cite...
 6.3.38: In 3040, construct an algebraic proof for the given statement. Cite...
 6.3.39: In 3040, construct an algebraic proof for the given statement. Cite...
 6.3.40: In 3040, construct an algebraic proof for the given statement. Cite...
 6.3.41: In 4143 simplify the given expression. Cite a property from Theorem...
 6.3.42: In 4143 simplify the given expression. Cite a property from Theorem...
 6.3.43: In 4143 simplify the given expression. Cite a property from Theorem...
 6.3.44: Consider the following set property: For all sets A and B, A B and ...
 6.3.45: Consider the following set property: For all sets A, B, and C, (A B...
 6.3.46: Let A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, and C = {5, 6, 7, 8}. Find ...
 6.3.47: Refer to the definition of symmetric difference given above. Prove ...
 6.3.48: Refer to the definition of symmetric difference given above. Prove ...
 6.3.49: Refer to the definition of symmetric difference given above. Prove ...
 6.3.50: Refer to the definition of symmetric difference given above. Prove ...
 6.3.51: Refer to the definition of symmetric difference given above. Prove ...
 6.3.52: Refer to the definition of symmetric difference given above. Prove ...
 6.3.53: Derive the set identity A (A B) = A from the properties listed in T...
 6.3.54: Derive the set identity A (A B) = A from the properties listed in T...
Solutions for Chapter 6.3: Disproofs, Algebraic Proofs, and Boolean Algebras
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 6.3: Disproofs, Algebraic Proofs, and Boolean Algebras
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 6.3: Disproofs, Algebraic Proofs, and Boolean Algebras includes 54 full stepbystep solutions. Discrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4. Since 54 problems in chapter 6.3: Disproofs, Algebraic Proofs, and Boolean Algebras have been answered, more than 57084 students have viewed full stepbystep solutions from this chapter.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

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.

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].

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.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

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.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

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

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

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.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).