 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 onetoone and onto function wi...
 7.3.15E: Suppose Y and Z are sets and g: Y ^ Z is a onetoone function. Thi...
 7.3.16E: If f: X ? Y and g: Y ? Z are functions and g ? f is onetoone, 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 onetoone, 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 onetoone 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
ISBN: 9780495391326
Solutions for Chapter 7.3
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 27 problems in chapter 7.3 have been answered, more than 43648 students have viewed full stepbystep solutions from this chapter. 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. Chapter 7.3 includes 27 full stepbystep solutions.

CayleyHamilton Theorem.
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. Blockdiagonal: 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).

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].

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.

Network.
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.

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 ().

Spanning set.
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)jI and det V = product of (Xk  Xi) for k > i.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.