 5.4.1: The accompanying figure represents a function. 5 6 4 7 8 8 9 10 11 ...
 5.4.2: The accompanying figure illustrates various binary relations from R...
 5.4.3: Using the equation f(x) = 2x 1 to describe the functional associati...
 5.4.4: Using the equation f(x) = x2 + 1 to describe the functional associa...
 5.4.5: If f: Z S Z is defined by f(x) = 3x, find f(A) for a. A = 51, 3, 56...
 5.4.6: If f: R S R is defined by f(x) = x2 , describe a. f(N). b. f(Z). c....
 5.4.7: The function f: 5all English words6 S Z. In each case, find f(S). a...
 5.4.8: The function f : 5binary strings6 S 5binary strings6. In each case,...
 5.4.9: True or false: a. An onto function means that every element in the ...
 5.4.10: True or false: a. If every element in the domain has an image, it m...
 5.4.11: Let S = 50, 2, 4, 66 and T = 51, 3, 5, 76. Determine whether each o...
 5.4.12: For any bijections in Exercise 11, describe the inverse function
 5.4.13: Let S = the set of all U. S. citizens alive today. Which of the fol...
 5.4.14: Let S = the set of people at a meeting, let T = the set of all shoe...
 5.4.15: Which of the following definitions describe functions from the doma...
 5.4.16: Which of the following definitions describe functions from the doma...
 5.4.17: Let f: R S R be defined by f(x) = xn , where n is a fixed, positive...
 5.4.18: Let f: R S R be defined by f(x) = n2x , where n is a fixed, positiv...
 5.4.19: Let A = 5x, y6 and let A* be the set of all strings of finite lengt...
 5.4.20: Let A = 5x, y6 and let A* be the set of all strings of finite lengt...
 5.4.21: Let A = 5x, y6 and let A* be the set of all strings of finite lengt...
 5.4.22: Let A = 5x, y6 and let A* be the set of all strings of finite lengt...
 5.4.23: Let P be the power set of 5a, b, c6. A function f: P S Z is defined...
 5.4.24: Let P be the power set of 5a, b6 and let S be the set of all binary...
 5.4.25: Let S = 5x 0 x [ R and x 16, and T = 5x 0 x [ R and 0 < x 16. Find ...
 5.4.26: Let S = 5a, b, c, d6 and T = 5x, y, z6. a. Give an example of a fun...
 5.4.27: Compute the following values. a. :3.4; b. <0.2= c. :0.5;
 5.4.28: Compute the following values. a. <51. 2= b. <5<1.2= = c. :2 * 3.7; ...
 5.4.29: What can be said about x if :x; =
 5.4.30: Prove that x= + 1 = <x + 1
 5.4.31: Prove that :x; = <x=.
 5.4.32: The ceiling function f(x) =
 5.4.33: Prove or disprove: a. < :x;= = x b. :2x; = 2:x;
 5.4.34: Prove or disprove: a. :x; + : y; = :x + y; b. :2x; = :x; + :x + 12;
 5.4.35: Prove that if 2k < n < 2k+1 then k = :log n; and k + 1 =
 5.4.36: Prove that if 2k n < 2k+1 then :log n; + 1 =
 5.4.37: Compute the value of the following expressions. a. 31 mod 11 b. 16 ...
 5.4.38: a. List five values x such that x mod 7 = 0. b. List five values x ...
 5.4.39: Prove or disprove: For any integers x and y, x mod 10 + y mod 10 = ...
 5.4.40: Prove that x y (mod n) if and only if x mod n = y mod n. (Recall th...
 5.4.41: Let S be a set and let A be a subset of S. The characteristic funct...
 5.4.42: Ackermanns function1 , mapping N2 to N, is a recursive function tha...
 5.4.43: Another rapidly growing function is the Smorynski function, which a...
 5.4.44: The Dwyer function also maps N2 to N and grows very rapidly, but it...
 5.4.45: Let S = 51, 2, 3, 46, T = 51, 2, 3, 4, 5, 66, and U = 56, 7, 8, 9, ...
 5.4.46: a. Let f: R S Z be defined by f(x) = :x;. Let g: Z S N be defined b...
 5.4.47: Let f : N S N be defined by f(x) = x + 1. Let g: N S N be defined b...
 5.4.48: The following functions map R to R. Give an equation describing the...
 5.4.49: Let f : S S T and g: T S U be functions. a. Prove that if g + f is ...
 5.4.50: a. Let f be a function, f : S S T. If there exists a function g: T ...
 5.4.51: For each of the following bijections f: R S R, find f 1 . a. f(x) =...
 5.4.52: Let f and g be bijections, f : S S T and g: T S U. Then f 1 and g1 ...
 5.4.53: Let A = 51, 2, 3, 4, 56. Write each of the following permutations o...
 5.4.54: Let A 5a, b, c, d6. Write each of the following permutations on A i...
 5.4.55: Let A be any set and let SA be the set of all permutations of A. Le...
 5.4.56: Find the composition of the following cycles representing permutati...
 5.4.57: Find the composition of the following cycles representing permutati...
 5.4.58: Find the composition of the following cycles representing permutati...
 5.4.59: Find the composition of the following cycles representing permutati...
 5.4.60: Find a permutation on an infinite set that cannot be written as a c...
 5.4.61: The function f written in cycle form as f = (4, 2, 8, 3) is a bijec...
 5.4.62: The pushdown store, or stack, is a storage structure that operates ...
 5.4.63: Let S = 52, 4, 6, 86 and T = 51, 5, 76. a. Find the number of funct...
 5.4.64: Let S = 5P, Q, R6 and T = 5k, l, m, n6. a. Find the number of funct...
 5.4.65: a. For 0S 0 = 2, 3, and 4, respectively, use the theorem on the num...
 5.4.66: Let A = 5a, b, c, d6. How many functions are in SA? How many of the...
 5.4.67: Let 0S 0 = n. In parts ae, find the number of a. functions from S ...
 5.4.68: a. A system development project calls for five different tasks to b...
 5.4.69: In a programming class of seven students, the instructor wants each...
 5.4.70: a. Find a calculus book and look up the Maclaurin series representa...
 5.4.71: Let f be a function, f : S S T. a. Define a binary relation r on S ...
 5.4.72: Prove that S(m, n), the number of ways to partition a set of m elem...
 5.4.73: By the definition of a function f from S to T, f is a subset of S T...
 5.4.74: Let f be a function, f: S S T. a. Show that for all subsets A and B...
 5.4.75: Let C be a collection of sets, and define a binary relation r on C ...
 5.4.76: Group the following sets into equivalence classes according to the ...
 5.4.77: a. Write a Scheme function to square a number. b. What is the outpu...
 5.4.78: Scheme also supports recursion, plus the usual control structures o...
Solutions for Chapter 5.4: Functions
Full solutions for Mathematical Structures for Computer Science  7th Edition
ISBN: 9781429215107
Solutions for Chapter 5.4: Functions
Get Full SolutionsSince 78 problems in chapter 5.4: Functions have been answered, more than 4175 students have viewed full stepbystep solutions from this chapter. Chapter 5.4: Functions includes 78 full stepbystep solutions. Mathematical Structures for Computer Science was written by Patricia and is associated to the ISBN: 9781429215107. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Mathematical Structures for Computer Science, edition: 7.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

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.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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).

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Iterative method.
A sequence of steps intended to approach the desired solution.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

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.

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).
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