- 22.214.171.124.1: Find the values of these expressions. a) 10 b) 1 + 1 c) 00 d) (1 + 0)
- 126.96.36.199.2: Find the values, if any, of the Boolean variable x that satisfy the...
- 188.8.131.52.3: a) Show that (1 . 1) + (0 . I + 0) = 1. b) Translate the equation i...
- 184.108.40.206.4: a) Show that (1 . 0) + (1 0) = 1. b) Translate the equation in part...
- 220.127.116.11.5: Use a table to express the values of each of these Boolean function...
- 18.104.22.168.6: Use a table to express the values of each of these Boolean function...
- 22.214.171.124.7: Use a 3-cube Q3 to represent each of the Boolean functions in Exerc...
- 126.96.36.199.8: Use a 3-cube Q3 to represent each of the Boolean functions in Exerc...
- 188.8.131.52.9: What values of the Boolean variables x and y satisfy xy =x+ y?
- 184.108.40.206.10: How many different Boolean functions are there of degree 7?
- 220.127.116.11.11: Prove the absorption law x + xy = x using the other laws in Table 5.
- 18.104.22.168.12: Show that F(x , y, z) = xy + xz + yz has the value 1 if and only if...
- 22.214.171.124.13: Show that xy + yz + xz = xy + yz + xz.
- 126.96.36.199.14: Verify the law of the double complement.
- 188.8.131.52.15: Verify the idempotent laws.
- 184.108.40.206.16: Verify the identity laws.
- 220.127.116.11.17: Verify the domination laws.
- 18.104.22.168.18: Verify the commutative laws.
- 22.214.171.124.19: Verify the associative laws.
- 126.96.36.199.20: Verify the first distributive law in Table 5.
- 188.8.131.52.21: Verify De Morgan's laws.
- 184.108.40.206.22: Verify the unit property.
- 220.127.116.11.23: Verify the zero property.
- 18.104.22.168.24: Simplify these expressions. a) x EB 0 b) x EB 1 c ) xEBx d ) xEBx
- 22.214.171.124.25: Show that these identities hold. a) x EBy = (x + y)(xy) b) x EB Y =...
- 126.96.36.199.26: Show that x EB y = y EB x.
- 188.8.131.52.27: Prove or disprove these equalities. a) x EB (y EBz) = (x EBy) EBz b...
- 184.108.40.206.28: Find the duals of these Boolean expressions. a) x + y b) xy c) xyz ...
- 220.127.116.11.29: Suppose that F is a Boolean function represented by a Boolean expre...
- 18.104.22.168.30: Show that if F and G are Boolean functions represented by Boolean e...
- 22.214.171.124.31: How many different Boolean functions F(x , y, z) are there such tha...
- 126.96.36.199.32: How many different Boolean functions F(x, y, z) are there such that...
- 188.8.131.52.33: Show that you obtain De Morgan's laws for propositions (in Table 5 ...
- 184.108.40.206.34: Show that you obtain the absorption laws for propositions (in Table...
- 220.127.116.11.35: tated properties hold in every Boolean algebra. 35. Show that in a ...
- 18.104.22.168.36: Show that in a Boolean algebra, every element x has a unique comple...
- 22.214.171.124.37: Show that in a Boolean algebra, the complement of the element 0 is ...
- 126.96.36.199.38: Prove that in a Boolean algebra, the law of the double complement h...
- 188.8.131.52.39: Show that De Morgan's laws hold in a Boolean algebra. That is, show...
- 184.108.40.206.40: Show that in a Boolean algebra, the modular properties hold. That i...
- 220.127.116.11.41: Show that in a Boolean algebra, if x v y = 0, then x = 0 and y = 0,...
- 18.104.22.168.42: Show that in a Boolean algebra, the dual of an identity, obtained b...
- 22.214.171.124.43: Show that a complemented, distributive lattice is a Boolean algebra.
Solutions for Chapter 11.1: Boolean Algebra
Full solutions for Discrete Mathematics and Its Applications | 6th Edition
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.
Upper triangular systems are solved in reverse order Xn to Xl.
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.
peA) = det(A - AI) has peA) = zero matrix.
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x - x) (x - x) T is positive (semi)definite; :E is diagonal if the Xi are independent.
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.
Dimension of vector space
dim(V) = number of vectors in any basis for V.
Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.
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.
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
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.
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.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.