 12.4.25E: Use the QuineMcCluskey method to simplify the sum ofproducts exp...
 12.4.1E: a) Draw a Kmap for a function in two variables and put a 1 in the ...
 12.4.2E: Find the sumofproducts expansions represented by each of these K...
 12.4.3E: Draw the Kmaps of these sumofproducts expansions in two variable...
 12.4.4E: Use a Kmap to find a minimal expansion as a Boolean sum of Boolean...
 12.4.5E: a) Draw a Kmap for a function in three variables. Put a 1 in the c...
 12.4.6E: Use Kmaps to find simpler circuits with the same output as each of...
 12.4.7E: Draw the Kmaps of these sumofproducts expansions in three variab...
 12.4.8E: Construct a Kmap for Use this Kmap to find the implicants, prime ...
 12.4.9E: Construct a Kmap for Use this Kmap to find the implicants, prime ...
 12.4.10E: Draw the 3cube Q3 and label each vertex with the minterm in the Bo...
 12.4.11E: Draw the 4cube Q4 and label each vertex with the minterm in the Bo...
 12.4.12E: Use a Kmap to find a minimal expansion as a Boolean sum of Boolean...
 12.4.13E: a) Draw a Kmap for a function in four variables.put a 1 in the cel...
 12.4.15E: Find the cells in a Kmap for Boolean functions with five variables...
 12.4.14E: Use a Kmap to find a minimal expansion as a Boolean sum of Boolean...
 12.4.16E: How many cells in a Kmap for Boolean functions with six variables ...
 12.4.18E: Show that cells in a Kmap for Boolean functions in five variables ...
 12.4.17E: a) How many cells does a Kmap in six variables have?______________...
 12.4.19E: Which rows and which columns of a 4 x 16 map for Boolean functions ...
 12.4.20E: Use Kmaps to find a minimal expansion as a Boolean sum of Boolean ...
 12.4.21E: Suppose that there are five members on a committee, but that Smith ...
 12.4.22E: Use the Quine—McCluskey method to simplify the sumofproducts expa...
 12.4.26E: Explain how Kmaps can be used to simplify productof sums expansi...
 12.4.24E: Use the QuineMcCluskey method to simplify the sum ofproducts exp...
 12.4.23E: Use the QuineMcCluskey method to simplify the sum ofproducts exp...
 12.4.28E: Draw a Kmap for the 16 minterms in four Boolean variables on the s...
 12.4.27E: Use the method from Exercise 26 to simplify the productofsums exp...
 12.4.29E: Build a circuit using OR gates, AND gates, and inverters that produ...
 12.4.30E: In Exercises 3032 find a minimal sumofproducts expansion, given ...
 12.4.31E: In Exercises 3032 find a minimal sumofproducts expansion, given ...
 12.4.32E: In Exercises 3032 find a minimal sumofproducts expansion, given ...
 12.4.33E: Show that products of k literals correspond to 2nk dimensional sub...
Solutions for Chapter 12.4: Discrete Mathematics and Its Applications 7th Edition
Full solutions for Discrete Mathematics and Its Applications  7th Edition
ISBN: 9780073383095
Solutions for Chapter 12.4
Get Full SolutionsSince 33 problems in chapter 12.4 have been answered, more than 219304 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 7. Chapter 12.4 includes 33 full stepbystep solutions. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095. This expansive textbook survival guide covers the following chapters and their solutions.

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.

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).

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

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 Fn1c can be computed with ne/2 multiplications. Revolutionary.

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

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Lucas numbers
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.

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).

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

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

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

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

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·