- 1.4.1: Consider the statements P: 5 is odd. Q: 23 + 1 is even. (a) Express...
- 1.4.2: Determine, with explanation, whether the statement 12 > 0 if and on...
- 1.4.3: For an integer n and the open sentences P(n): 3n2 is even. Q(n): n3...
- 1.4.4: For an integer n and the open sentences P(n): n is odd. Q(n): n2 is...
- 1.4.5: For an integer n, consider the biconditional 2n > n2 if and only if...
- 1.4.6: For an integer n, consider the open sentences P(n): 5n + 7 is even....
- 1.4.7: For integers a and b, consider the biconditional ab is even if and ...
- 1.4.8: For integers a and b, consider the biconditional a + b is even if a...
- 1.4.9: For integers a, b and c, consider the biconditional At least two of...
- 1.4.10: For an integer n, consider the open sentences P(n) : n(n + 1)(2n + ...
- 1.4.11: For two statements P and Q, give an explanation why the bicondition...
- 1.4.12: For every two statements P and Q, use truth tables to verify the fo...
Solutions for Chapter 1.4: Biconditionals
Full solutions for Discrete Mathematics | 1st Edition
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
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.
Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
lA-II = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).
Every v in V is orthogonal to every w in W.
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).
Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.
Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.
Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.