 2.3.1E: Why is / not a function from R to R if
 2.3.2E: Determine whether f is a function from Z to R if
 2.3.3E: Determine whether f is a function from the set of all bit strings t...
 2.3.4E: Find the domain and range of these functions. Note that in each cas...
 2.3.5E: Find the domain and range of these functions. Note that in each eas...
 2.3.6E: Find the domain and range of these functions.a) the function that a...
 2.3.7E: Find the domain and range of these functions.a) the function that a...
 2.3.8E: Find these values.
 2.3.9E: Find these values.
 2.3.10E: Determine whether each of these functions from [a, b, c, d] to itse...
 2.3.11E: Which functions in Exercise 10 are onto?
 2.3.12E: Determine whether each of these functions from Z to Z is onetoone.
 2.3.13E: Which functions in Exercise 12 are onto?
 2.3.14E: Determine whether f: Z x Z ? Z is onto if
 2.3.15E: Determine whether the function f: Z x Z ? Z is onto if
 2.3.16E: Consider these functions from the set of students in a discrete mat...
 2.3.17E: Consider these functions from the set of teachers in a school. Unde...
 2.3.18E: Specify a codomain for each of the functions in Exercise 16. Under ...
 2.3.19E: Specify a codomain for each of the functions in Exercise 17. Under ...
 2.3.20E: Give an example of a function from N to N that isa) onetoone but ...
 2.3.21E: an explicit formula for a function from the set of integers to the ...
 2.3.22E: Determine whether each of these functions is a bijection from R to R.
 2.3.23E: Determine whether each of these functions is a bijection from R to R
 2.3.24E: Let f: R ? R and let f(x) > 0 for all .x ? R. Show that f(x) is str...
 2.3.25E: Let f: R ? R and let f(x) > 0 for all .x ? R. Show f(x) is strictly...
 2.3.26E: a) Prove that a strictly increasing function from R to it self is o...
 2.3.27E: a) Prove that a strictly decreasing function from R to itself is on...
 2.3.28E: Show that the function f(x) = ex from the set of real numbers to th...
 2.3.29E: Show that the function f(x) = x from the set of real numbers to t...
 2.3.30E: Let S= {?1, 0, 2, 4, 7}. Find f(S) if
 2.3.31E: Let f(x) = [x2/3]. Find f(S) if
 2.3.32E: Let f(x) = 2x where the domain is the set of real numbers. What is
 2.3.33E: Suppose that g is a function from A to B and f is a function from B...
 2.3.34E: If f and f o g are onetoone. does it follow that g is onetoone?...
 2.3.35E: If f and f o g are onto, does it follow that g is onto? Justify you...
 2.3.36E: Find f o g and g o f, where f(x) — x2 + 1 and g(x) = .x + 2. are fu...
 2.3.37E: Find f + g and fg for the functions f and g given in Exercise 36.
 2.3.38E: Let f(x) = ax + b and g(x) = cx + J. where a, b, c. and d arc const...
 2.3.39E: Show that the function f(x) = ax + b from R to R is invertible. whe...
 2.3.40E: Let f be a function from the set A to the set B. Let S and T be sub...
 2.3.41E: a) Give an example to show that the inclusion in part (b) in Exerci...
 2.3.42E: Let f be the function from R to R defined by f(x) = x2. Find
 2.3.43E: Let g(x) = [x]Find
 2.3.44E: Let f be a function from A to B. Let S and T be subsets of B. Show ...
 2.3.45E: Let f be a function from the set A to the set B. Let S be a subset ...
 2.3.46E: Show that is the closest integer to the number .x, except when x is...
 2.3.47E: Show that is the closest integer to the number x, except when x is ...
 2.3.48E: Show that if x is a real number, then if x is not an integer and if...
 2.3.49E: Show that if x is a real number, then
 2.3.50E: Show that if x is a real number and m is an integer, then:
 2.3.51E: Show that if x is a real number and n is an integer, then
 2.3.52E: Show that if x is a real number and n is an integer, then
 2.3.53E: Prove that if n is an integer, then [n/2]= n/2 if n is even and (n ...
 2.3.54E: Prove that if x is a real number, then and
 2.3.55E: The function INT is found on some calculators, where INT(x) = when ...
 2.3.56E: Let a and b be real numbers with a < b. Use the floor and or ceilin...
 2.3.57E: Let a and b be real numbers with a<b. Use the floor and/or ceiling ...
 2.3.58E: How many bytes are required to encode n bits of data where n equals...
 2.3.59E: How many bytes are required to encode n bits of data where n equals...
 2.3.60E: How many ATM cells (described in Example 28) can be transmitted in ...
 2.3.61E: Data are transmitted over a particular Ethernet network in blocks o...
 2.3.62E: Draw the graph of the function f(n) = 1 — n2 from Z to Z.
 2.3.63E: Draw the graph of the function f(x)= [2x] from R to R.
 2.3.64E: Draw the graph of the function f(x) = [x/2] from R to R.
 2.3.65E: Draw the graph of the function f(x) =[x] + [x/2] from R to R.
 2.3.66E: Draw the graph of the function f(x) = [x] + [x/2] from R to R.
 2.3.67E: Draw graphs of each of these functions.
 2.3.68E: Draw graphs of each of these functions.
 2.3.69E: Find the inverse function of f(x) = x 3 + 1.
 2.3.70E: Suppose that S is an invertible function from Y to Z and g is an in...
 2.3.71E: Let S be a subset of a universal set U. The characteristic function...
 2.3.72E: Suppose that f is a function from A to B. where A and B are finite ...
 2.3.73E: Prove or disprove each of these statements about the floor and ceil...
 2.3.74E: Prove or disprove each of these statements about the floor and ceil...
 2.3.75E: Prove that ifa is a positive real number, then
 2.3.76E: Let x be a real number. Show that [3x] =
 2.3.77E: For each of these partial functions, determine its domain, codomain...
 2.3.78E: a) Show that a partial function from .A to B can be viewed as a fun...
 2.3.79E: a) Show that if a set S has cardinality m, where m is a positive in...
 2.3.80E: Show that a set S is infinite if and only if there is a proper subs...
Solutions for Chapter 2.3: Discrete Mathematics and Its Applications 7th Edition
Full solutions for Discrete Mathematics and Its Applications  7th Edition
ISBN: 9780073383095
Solutions for Chapter 2.3
Get Full SolutionsChapter 2.3 includes 80 full stepbystep solutions. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095. Since 80 problems in chapter 2.3 have been answered, more than 255600 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: 7.

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.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

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.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

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

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).