 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: 4th. Since 54 problems in chapter 6.3: Disproofs, Algebraic Proofs, and Boolean Algebras have been answered, more than 27566 students have viewed full stepbystep solutions from this chapter.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

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

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

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.

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

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.

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.

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

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

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 .

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).

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).

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

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

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

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

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

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