 1.2.1: Identify the antecedent and the consequent for each of the followin...
 1.2.2: . Write the converse and contrapositive of each conditional sentenc...
 1.2.3: What can be said about the truth value of Q when(a) P is false and ...
 1.2.4: dentify the antecedent and consequent for each conditional sentence...
 1.2.5: Which of the following conditional sentences are true? (a) If trian...
 1.2.6: . Which of the following are true? (a) Triangles have three sides s...
 1.2.7: Make truth tables for these propositional forms.(a) (b) (c) (d)(e)(...
 1.2.8: Prove Theorem 1.2.2 by constructing truth tables for each equivalence.
 1.2.9: Determine whether each statement qualifies as a definition.(a) is a...
 1.2.10: Rewrite each of the following sentences using logical connectives. ...
 1.2.11: . Dictionaries indicate that the conditional meaning of unless is p...
 1.2.12: Show that the following pairs of statements are equivalent.(a) and ...
 1.2.13: . Give, if possible, an example of a true conditional sentence for ...
 1.2.14: Give, if possible, an example of a false conditional sentence for w...
 1.2.15: Give the converse and contrapositive of each sentence of Exercises ...
 1.2.16: Determine whether each of the following is a tautology, a contradic...
 1.2.17: The inverse, or opposite, of the conditional sentence(a) Show that ...
Solutions for Chapter 1.2: Conditionals and Biconditionals
Full solutions for A Transition to Advanced Mathematics  7th Edition
ISBN: 9780495562023
Solutions for Chapter 1.2: Conditionals and Biconditionals
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Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

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

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

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).

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

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

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

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

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

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

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).