 2.3.2.3.1: Why is f not a function from R to R if a) f(x) = 1/x? b) f(x) = .JX...
 2.3.2.3.2: Determine whether f is a function from Z to R if a) f(n) = n. b) f(...
 2.3.2.3.3: Determine whether f is a function from the set of all bit strings t...
 2.3.2.3.4: Find the domain and range of these functions. Note that in each cas...
 2.3.2.3.5: Find the domain and range of these functions. Note that in each cas...
 2.3.2.3.6: Find the domain and range of these functions. a) the function that ...
 2.3.2.3.7: Find the domain and range of these functions. a) the function that ...
 2.3.2.3.8: Find these values. a) Ll .1J c) L O. lJ e) r2.991 g) L! + r!l J b)...
 2.3.2.3.9: Find these values. a) rl b) LJ c) rl d) LJ e) r31 1) L lJ g) L! ...
 2.3.2.3.10: Determine whether each of these functions from {a, b, c, d} to itse...
 2.3.2.3.11: Which functions in Exercise 10 are onto?
 2.3.2.3.12: Which functions in Exercise 10 are onto?
 2.3.2.3.13: Which functions in Exercise 12 are onto?
 2.3.2.3.14: Determine whether f: Z x Z Z is onto if a) f(m, n) =2m n. b) f(m, ...
 2.3.2.3.15: Detennine whether the function f: Z x Z Z is onto if a) f(m, n) = m...
 2.3.2.3.16: Give an example of a function from N to N that is a) onetoone but...
 2.3.2.3.17: Give an explicit fonnula for a function from the set of integers to...
 2.3.2.3.18: Detennine whether each of these functions is a bijection from R to ...
 2.3.2.3.19: Detennine whether each of these functions is a bijection from R to ...
 2.3.2.3.20: Let f: R R and let f(x) > 0 for all x E R. Show that f(x) is strict...
 2.3.2.3.21: Letf: R R and letf(x) > O. Showthatf(x) is strictly decreasing if a...
 2.3.2.3.22: Give an example of an increasing function with the set of real numb...
 2.3.2.3.23: Give an example of a decreasing function with the set of real numbe...
 2.3.2.3.24: Show that the function f(x) = eX from the set of real number to the...
 2.3.2.3.25: Show that the function f(x) = Ix I from the set of real numbers to ...
 2.3.2.3.26: Let S = {I , 0, 2, 4, 7}. Find f(S) if a) f(x) = 1. b) f(x) = 2x +...
 2.3.2.3.27: Let f(x) = Lx2 /3J . Find f(S) if a) S = {2,  1 , 0, 1 , 2, 3}. b...
 2.3.2.3.28: Let f(x) = 2x. What is a) f(Z)? b) f(N)? c) f(R)?
 2.3.2.3.29: Suppose that g is a function from A to B and f is a function from B...
 2.3.2.3.30: If f and fog are onetoone, does it follow that g is onetoone? J...
 2.3.2.3.31: If f and fog are onto, does it follow that g is onto? Justify your ...
 2.3.2.3.32: Find fog and g 0 f, where f(x) = x2 + 1 and g(x) = x + 2, are funct...
 2.3.2.3.33: Find f + g and fg for the functions f and g given in Exercise 32.
 2.3.2.3.34: Letf(x) = ax + b andg(x) = ex + d, where a, b, e, and d are constan...
 2.3.2.3.35: Show that the function f(x) = ax + b from R to R is invertible, whe...
 2.3.2.3.36: Let f be a function from the set A to the set B. Let S and T be sub...
 2.3.2.3.37: Give an example to show that the inclusion in part (b) in Exercise ...
 2.3.2.3.38: Let f be the function from R to R defined by f(x) = x2 Find a) fI ...
 2.3.2.3.39: Let g(x) = LxJ . Find a) g I ({O}). b) gI ({ I , O, I }). c) g ...
 2.3.2.3.40: Let f be a function from A to B. Let S and T be subsets of B. Show ...
 2.3.2.3.41: Let f be a function from A to B. Let S be a subset of B. Show that ...
 2.3.2.3.42: Show that Lx + J is the closest integer to the number x, except whe...
 2.3.2.3.43: Show that f x  1 is the closest integer to the number x, except wh...
 2.3.2.3.44: Show that if x is a real number, then fxl  LxJ = I if x is not an ...
 2.3.2.3.45: Show that if x is a real number, then xI < Lx J ::::: x ::::: fxl ...
 2.3.2.3.46: Show that if x is a real number and m is an integer, then fx + ml =...
 2.3.2.3.47: Show that if x is a real number and n is an integer, then a) x < n ...
 2.3.2.3.48: Show that if x is a real number and n is an integer, then a) x ::::...
 2.3.2.3.49: Prove that ifn is an integer, then Ln/2J = n/2 ifn is even and (n ...
 2.3.2.3.50: Prove that if x is a real number, then L x J =  f x 1 and f xl =...
 2.3.2.3.51: The function INT is found on some calculators, where INT(x) = Lx J ...
 2.3.2.3.52: Let a and b be real numbers with a < b. Use the floor and/or ceilin...
 2.3.2.3.53: Let a and b be real numbers with a < b. Use the floor and/or ceilin...
 2.3.2.3.54: How many bytes are required to encode n bits of data where n equals...
 2.3.2.3.55: How many bytes are required to encode n bits of data where n equals...
 2.3.2.3.56: How many ATM cells (described in Example 26) can be transmitted in ...
 2.3.2.3.57: Data are transmitted over a particular Ethernet network in blocks o...
 2.3.2.3.58: Draw the graph of the function fen) = 1  n2 from Z to z.
 2.3.2.3.59: Draw the graph of the function f(x) = L2xJ from R to R.
 2.3.2.3.60: Draw the graph of the function f(x) = Lx/2J from R to R.
 2.3.2.3.61: Draw the graph of the function f(x) = Lx J + Lx /2J from R to R.
 2.3.2.3.62: Draw the graph ofthe function f(x) = fxl + Lx/2J from R to R.
 2.3.2.3.63: Draw graphs of each of these functions. a) f(x) = Lx + J b) f(x) = ...
 2.3.2.3.64: Draw graphs of each of these functions. a) f(x) = f3x  21 b) f(x) ...
 2.3.2.3.65: Find the inverse function of fex) = x3 + 1.
 2.3.2.3.66: Suppose that f is an invertible function from Y to Z and g is an in...
 2.3.2.3.67: Let S be a subset of a universal set U. The characteristic function...
 2.3.2.3.68: Suppose that f is a function from A to B, where A and B are finite ...
 2.3.2.3.69: Prove or disprove each ofthese statements about the floor and ceili...
 2.3.2.3.70: Prove or disprove each ofthese statements about the floor and ceili...
 2.3.2.3.71: Prove that if x is a positive real number, then a) LvTxT J = L..jX ...
 2.3.2.3.72: Let x be a real number. Show that L3xJ = LxJ + Lx + 1 J + Lx + J.
 2.3.2.3.73: For each of these partial functions, determine its domain, codomain...
 2.3.2.3.74: a) Show that a partial function from A to B can be viewed as a func...
 2.3.2.3.75: a) Show that if a set S has cardinality m, where m is a positive in...
 2.3.2.3.76: Show that a set S is infinite if and only ifthere is a proper subse...
 2.3.2.3.77: Show that the polynomial function f : Z+ x Z+ + Z+ with f(m, n) =...
 2.3.2.3.78: Show that when you substitute (3n + 1)2 for each occurrence of n an...
Solutions for Chapter 2.3: Basic Structures: Sets, Functions, Sequences, and Sums
Full solutions for Discrete Mathematics and Its Applications  6th Edition
ISBN: 9780073229720
Solutions for Chapter 2.3: Basic Structures: Sets, Functions, Sequences, and Sums
Get Full SolutionsSince 78 problems in chapter 2.3: Basic Structures: Sets, Functions, Sequences, and Sums have been answered, more than 34151 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 6. Chapter 2.3: Basic Structures: Sets, Functions, Sequences, and Sums includes 78 full stepbystep solutions. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073229720.

Block matrix.
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.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of 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.

Condition number
cond(A) = c(A) = IIAIlIIAIII = 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.

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].

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

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 nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

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.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Multiplier eij.
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).

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

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.