- Chapter 5.29.16: Let f D f.1; 2/; .2; 3/; .3; 4/g and g D f.2; 1/; .3; 1/; .4; 2/g. ...
- Chapter 5.29.17: Let f .x/ D ax C b where a 6D 0. Find f 1 .x/.
- Chapter 5.29.18: Find all functions of the form f .x/ D ax C b such that .f f /.x/ D...
- Chapter 5.29.19: Suppose A and B are sets and f is a function with f W A ! B. Suppos...
- Chapter 5.29.20: Let A D f1; 2; 3g and let B D f3; 4; 5; 6g. a. How many functions f...
- Chapter 5.29.21: Let A and B be n-element sets. How many functions are there from th...
- Chapter 5.29.22: Suppose f W A ! B is one-to-one and g W B ! A is one-to-one. Must i...
- Chapter 5.29.23: Let f W Z ! N by f .x/ D jxj. a. Is f one-to-one? b. Is f onto? Pro...
- Chapter 5.29.24: Let f W Z ! Z by f .x/ D x 3 . a. Is f one-to-one? b. Is f onto? Pr...
- Chapter 5.29.25: Functions are relations, although it is not customary to consider w...
- Chapter 5.29.26: The squares of a 9 9 chess board are arbitrarily colored black and ...
- Chapter 5.29.27: Let A D f1; 2; 3; 4; 5g with f W A ! A, g W A ! A, and h W A ! A. W...
- Chapter 5.29.28: Suppose f; g W R ! R are defined by f .x/ D x 2 C x 1 and g.x/ D 3x...
- Chapter 5.29.29: Let f; g; h W R ! R defined by f .x/ D 3x 4, g.x/ D ax C b, and h.x...
- Chapter 5.29.30: In Exercise 9.14 you were asked to evaluate 0 0 . (The answer is 0 ...
- Chapter 5.29.31: Let A be a set. Suppose f and g are functions f W A ! A and g W A !...
- Chapter 5.29.32: Let be a permutation of f1; 2; 3; : : : ; 9g defined by the 29 arra...
- Chapter 5.29.33: Let n be a positive integer and let 2 Sn. Prove there is a positive...
- Chapter 5.29.34: Let n be a positive integer and ; 2 Sn. Evaluate Xn kD1 .k/ .k/ and...
- Chapter 5.29.35: Let n be a positive integer and let 2 Sn. a. Prove that can be writ...
- Chapter 5.29.36: Let n be a positive integer and 2 Sn. Let x1; x2; : : : ; xn be rea...
- Chapter 5.29.37: Let T be a tetrahedron (a solid figure with four triangular faces) ...
- Chapter 5.29.38: Let x be a real number and suppose that bxc D dxe. What can you con...
- Chapter 5.29.39: Show that 2 n is O.3n /, but 3 n is not O.2n /.
Solutions for Chapter Chapter 5: Functions
Full solutions for Mathematics: A Discrete Introduction | 3rd Edition
Upper triangular systems are solved in reverse order Xn to Xl.
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
A = CTC = (L.J]))(L.J]))T for positive definite A.
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.
cond(A) = c(A) = IIAIlIIA-III = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
Invert A by row operations on [A I] to reach [I A-I].
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
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.
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
Every v in V is orthogonal to every w in W.
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.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·
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.