- 11.3.1: It was proved in Theorem 11.63 that for every two elements a and b ...
- 11.3.2: What property is dual to each of the following properties? (a) x + ...
- 11.3.3: In the Boolean algebra B, compute the following: (a) 1 0 + (1 + 0)....
- 11.3.4: Let x and y be Boolean variables. Prove that x (x +y) = x y (a) by ...
- 11.3.5: Simplify each of the following Boolean expressions. (a) (w + x) (y ...
- 11.3.6: In (a)-(e), find the values of the Boolean expressions for x = 1, y...
- 11.3.7: Construct a table that displays the values of each of the following...
- 11.3.8: The values of three Boolean functions f1, f2 and f3 of degree 3 are...
- 11.3.9: Find the sum-of-products expansion of the Boolean function f of deg...
- 11.3.10: The values of two Boolean functions f and g of degree 3 are given i...
- 11.3.11: The Boolean function f(x, y, z) = (x + y + z)(x + y + z) is express...
- 11.3.12: The Boolean function f(x, y, z) = xyz +xyz +xyz is expressed in the...
- 11.3.13: Write the Boolean expression that represents the combinatorial circ...
- 11.3.14: Write the Boolean expression that represents the combinatorial circ...
- 11.3.15: Write the Boolean expression that represents the combinatorial circ...
- 11.3.16: The exclusive-OR, denoted by XOR, of two Boolean variables x and y ...
Solutions for Chapter 11.3: Boolean Algebras
Full solutions for Discrete Mathematics | 1st Edition
Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.
A = CTC = (L.J]))(L.J]))T for positive definite A.
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.
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.
Invert A by row operations on [A I] to reach [I A-I].
Gram-Schmidt 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.
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = 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 B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.
Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.
Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
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.
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.