- 12.R.1RQ: Define a Boolean function of degree n.
- 12.R.2RQ: How many Boolean functions of degree two are there?
- 12.R.3RQ: Give a recursive definition of the set of Boolean expressions.
- 12.R.4RQ: a) What is the dual of a Boolean expression?________________b) What...
- 12.R.5RQ: Explain how to construct the sum-of-products expansion of a Boolean...
- 12.R.7RQ: Explain how to build a circuit for a light controlled by two switch...
- 12.R.6RQ: a) What does it mean for a set of operators to be functionally comp...
- 12.R.8RQ: Construct a half adder using OR gates, AND gates, and inverters.
- 12.R.9RQ: Is there a single type of logic gate that can be used to build all ...
- 12.R.10RQ: a) Explain how K-maps can be used to simplify sum-of-products expan...
- 12.R.11RQ: a) Explain how K-maps can be used to simplify sum-of-products expan...
- 12.R.12RQ: a) What is a don’t care condition?________________b) Explain how do...
- 12.R.13RQ: a) Explain how to use the Quine-McCluskey method to simplify sum-of...
Solutions for Chapter 12.R: Discrete Mathematics and Its Applications 7th Edition
Full solutions for Discrete Mathematics and Its Applications | 7th 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.
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
Column space C (A) =
space of all combinations of the columns of A.
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 n-l - An).
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.
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.
Incidence matrix of a directed graph.
The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .
A symmetric matrix with eigenvalues of both signs (+ and - ).
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.
Outer product uv T
= column times row = rank one matrix.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
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}.