- 7.3.1E: In, functions f and g are defined by arrow diagrams. Find g ? f and...
- 7.3.2E: In, functions f and g are defined by arrow diagrams. Find g ? f and...
- 7.3.3E: In, functions F and G are defined by formulas. Find G ? F and F ? G...
- 7.3.4E: In, functions F and G are defined by formulas. Find G ? F and F ? G...
- 7.3.5E: Define f: R ? R by the rule f(x) = ?x for all real numbers x. Find ...
- 7.3.6E: Define F: Z ? Z and G: Z ? Z by the rules F(a) = 7a and G(a) = amod...
- 7.3.7E: Define H: Z ? Z and K: Z ? Z by the rules H(a) = 6a and K(a) = amod...
- 7.3.8E: Define L: Z ? Z and M: Z ? Z by the rules L(a) = a2 and M(a) = a mo...
- 7.3.9E: The functions of each pair are inverse to each other. For each pair...
- 7.3.10E: The functions of each pair are inverse to each other. For each pair...
- 7.3.11E: The functions of each pair are inverse to each other. For each pair...
- 7.3.12E: Explain how it follows from the definition of logarithm thata. logb...
- 7.3.13E: Prove Theorem (b): If f is any function from a set X to a set Y, th...
- 7.3.14E: Prove Theorem (b): If f: X ? Y is a one-to-one and onto function wi...
- 7.3.15E: Suppose Y and Z are sets and g: Y ^ Z is a one-to-one function. Thi...
- 7.3.16E: If f: X ? Y and g: Y ? Z are functions and g ? f is one-to-one, mus...
- 7.3.17E: If f: X ? Y and g: Y ? Z are functions and g ? f is onto, must f be...
- 7.3.18E: If f: X ? Y and g: Y ? Z are functions and g ? f is one-to-one, mus...
- 7.3.19E: If f: X ? Y and g: Y ? Z are functions and g ? f is onto, must g be...
- 7.3.20E: Let f: W ? X, g: X ? Y, and h: Y ? Z be functions. Must h ? (g ? f)...
- 7.3.21E: True or False? Given any set X and given any functions f: X ? X, g:...
- 7.3.22E: True or False? Given any set X and given any functions f: X ? X, g:...
- 7.3.23E: Find g ? f, (g ? f)?1, g?1, f?1, and f?1 ? g?1, and state how (g ? ...
- 7.3.24E: Find g ? f, (g ? f)?1, g?1, f?1, and f?1 ? g?1, and state how (g ? ...
- 7.3.25E: Prove or give a counterexample: If f: X ? Y and g: Y ? X are functi...
- 7.3.26E: Suppose f: X ? Y and g: Y ? Z are both one-to-one and onto. Prove t...
- 7.3.27E: Let f: X ? Y and g: Y ? Z. Is the following property true or false?...
Solutions for Chapter 7.3: Discrete Mathematics with Applications 4th Edition
Full solutions for Discrete Mathematics with Applications | 4th Edition
peA) = det(A - AI) has peA) = zero matrix.
Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).
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].
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 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.
Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.
Left inverse A+.
If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.
A directed graph that has constants Cl, ... , Cm associated with the edges.
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.
R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.