 2.2.1E: Rewrite the statements in in ifthen form.ExerciseThis loop will re...
 2.2.2E: Rewrite the statements in in ifthen form.ExerciseI am on time for ...
 2.2.3E: Rewrite the statements in in ifthen form.ExerciseFreeze or I’ll sh...
 2.2.4E: Rewrite the statements in in ifthen form.ExerciseFix my ceiling or...
 2.2.5E: Construct truth tables for the statement forms.Exercise
 2.2.6E: Construct truth tables for the statement forms.Exercise
 2.2.7E: Construct truth tables for the statement forms.Exercise
 2.2.8E: Construct truth tables for the statement forms.Exercise
 2.2.9E: Construct truth tables for the statement forms.Exercise
 2.2.10E: Construct truth tables for the statement forms.Exercise
 2.2.11E: Construct truth tables for the statement forms.Exercise
 2.2.12E: Use the logical equivalence established in Example, to rewrite the ...
 2.2.13E: Use truth tables to verify the following logical equivalences. Incl...
 2.2.14E: a. Show that the following statement forms are all logically equiva...
 2.2.15E: Determine whether the following statement forms are logically equiv...
 2.2.16E: Write each of the two statements in symbolic form and determine whe...
 2.2.17E: Write each of the two statements in symbolic form and determine whe...
 2.2.18E: Write each of the following three statements in symbolic form and d...
 2.2.19E: True or false? The negation of “If Sue is Luiz’s mother, then Ali i...
 2.2.20E: Write negations for each of the following statements. (Assume that ...
 2.2.21E: Suppose that p and q are statements so that p ^ q is false. Find th...
 2.2.22E: Write contrapositives for the statements of exercise.ExerciseWrite ...
 2.2.23E: Write the converse and inverse for each statement of exercise.Write...
 2.2.24E: Use truth tables to establish the truth of each statement.ExerciseA...
 2.2.25E: Use truth tables to establish the truth of each statement.ExerciseA...
 2.2.26E: Use truth tables to establish the truth of each statement.ExerciseA...
 2.2.27E: Use truth tables to establish the truth of each statement.ExerciseT...
 2.2.28E: “Do you mean that you think you can find out the answer to it?” sai...
 2.2.29E: If statement forms P and Q are logically equivalent, then P ? Q is ...
 2.2.30E: If statement forms P and Q are logically equivalent, then P ? Q is ...
 2.2.31E: If statement forms P and Q are logically equivalent, then P ? Q is ...
 2.2.32E: Rewrite each of the statements as a conjunction of two ifthen stat...
 2.2.33E: Rewrite each of the statements as a conjunction of two ifthen stat...
 2.2.34E: Rewrite the statements in ifthen form in two ways, one of which is...
 2.2.35E: Rewrite the statements in ifthen form in two ways, one of which is...
 2.2.36E: Taking the long view on your education, you go to the Prestige Corp...
 2.2.37E: In, rewrite the statements in ifthen form.ExercisePayment will be ...
 2.2.38E: In, rewrite the statements in ifthen form.ExerciseAnn will go unle...
 2.2.39E: In, rewrite the statements in ifthen form.ExerciseThis door will n...
 2.2.40E: Rewrite the statements in ifthen form.ExerciseCatching the 8:05 bu...
 2.2.41E: Rewrite the statements in ifthen form.ExerciseHaving two 45° angle...
 2.2.42E: Use the contrapositive to rewrite the statements in ifthen form in...
 2.2.43E: Use the contrapositive to rewrite the statements in ifthen form in...
 2.2.44E: Note that “a sufficient condition for s is r” means r is a sufficie...
 2.2.45E: Note that “a sufficient condition for s is r” means r is a sufficie...
 2.2.46E: “If compound X is boiling, then its temperature must be at least 15...
 2.2.47E: In 47–50 (a) use the logical equivalences p ?q ??p ? q and p ? q ? ...
 2.2.48E: In 47–50 (a) use the logical equivalences p ?q ??p ? q and p ? q ? ...
 2.2.49E: In 47–50 (a) use the logical equivalences p ?q ??p ? q and p ? q ? ...
 2.2.50E: In 47–50 (a) use the logical equivalences p ?q ??p ? q and p ? q ? ...
 2.2.51E: Given any statement form, is it possible to find a logically equiva...
Solutions for Chapter 2.2: Discrete Mathematics with Applications 4th Edition
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 2.2
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 51 problems in chapter 2.2 have been answered, more than 36776 students have viewed full stepbystep solutions from this chapter. Chapter 2.2 includes 51 full stepbystep solutions. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4. Discrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

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.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

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.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

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.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.