 8.1.1: Let B = 50, 1, a, a6, and let + and # be binary operations on B. Th...
 8.1.2: a. What does the universal bound property (Practice 3) become in th...
 8.1.3: Define two binary operations + and # on the set Z of integers by x ...
 8.1.4: Let M2(Z) denote the set of 2 2 matrices with integer entries, and ...
 8.1.5: Let S be the set 50, 16. Then S2 is the set of all ordered pairs of...
 8.1.6: Let n be a positive integer whose decomposition into prime factors ...
 8.1.7: Prove the following property of Boolean algebras. Give a reason for...
 8.1.8: Prove the following property of Boolean algebras. Give a reason for...
 8.1.9: Prove the following properties of Boolean algebras. Give a reason f...
 8.1.10: Prove the following properties of Boolean algebras. Give a reason f...
 8.1.11: Prove the following properties of Boolean algebras. Give a reason f...
 8.1.12: Prove the following properties of Boolean algebras. Give a reason f...
 8.1.13: Prove that in any Boolean algebra, x # y + x # y = y if and only if...
 8.1.14: Prove that in any Boolean algebra, x # y = 0 if and only if x # y = x.
 8.1.15: A new binary operation ! in a Boolean algebra (exclusive OR) is def...
 8.1.16: Prove that for any Boolean algebra: a. If x + y = 0, then x = 0 and...
 8.1.17: Prove that the 0 element in any Boolean algebra is unique; prove th...
 8.1.18: a. Find an example of a Boolean algebra with elements x, y, and z f...
 8.1.19: Let (S, d) and (S, d) be two partially ordered sets. (S, d) is isom...
 8.1.20: Find an example of two partially ordered sets (S, d) and (S, d) and...
 8.1.21: Let S = 50, 16 and let a binary operation # be defined on S by # 0 ...
 8.1.22: Consider the fourelement Boolean algebra defined in Exercise 6 wit...
 8.1.23: Let R denote the real numbers and R+ the positive real numbers. Add...
 8.1.24: An isomorphism from the Boolean algebra with set B = 50, 1, a, a6 t...
 8.1.25: Consider the set B of all functions mapping 50, 162 to 50, 16. We c...
 8.1.26: Let P, Q, and R be three statements in propositional logic with sta...
 8.1.27: Suppose that 3B, +, # , , 0, 14 and 3b, &, *, , f, 14 are isomorphi...
 8.1.28: According to the theorem on finite Boolean algebras, which we did n...
 8.1.29: A Boolean algebra may also be defined as a partially ordered set wi...
 8.1.30: a. Let n be a positive integer and consider B to be the set of all ...
Solutions for Chapter 8.1: Boolean Algebra Structure
Full solutions for Mathematical Structures for Computer Science  7th Edition
ISBN: 9781429215107
Solutions for Chapter 8.1: Boolean Algebra Structure
Get Full SolutionsChapter 8.1: Boolean Algebra Structure includes 30 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 30 problems in chapter 8.1: Boolean Algebra Structure have been answered, more than 21202 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Mathematical Structures for Computer Science, edition: 7. Mathematical Structures for Computer Science was written by and is associated to the ISBN: 9781429215107.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

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.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

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).

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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 II· Condition numbers measure the sensitivity of the output to change in the input.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Iterative method.
A sequence of steps intended to approach the desired solution.

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).

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

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.

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

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

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

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.