 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 sumofproducts 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 Kmaps can be used to simplify sumofproducts expan...
 12.R.11RQ: a) Explain how Kmaps can be used to simplify sumofproducts 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 QuineMcCluskey method to simplify sumof...
Solutions for Chapter 12.R: Discrete Mathematics and Its Applications 7th Edition
Full solutions for Discrete Mathematics and Its Applications  7th Edition
ISBN: 9780073383095
Solutions for Chapter 12.R
Get Full SolutionsDiscrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095. Chapter 12.R includes 13 full stepbystep solutions. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 7. Since 13 problems in chapter 12.R have been answered, more than 218458 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

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.

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.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: 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.

Factorization
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 edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Norm
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.

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

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Stiffness 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}.