 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 116038 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.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

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).

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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 .

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.

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

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

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

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.