- 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 one-to-one.
- 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) one-to-one 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 one-to-one. does it follow that g is one-to-one?...
- 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
Chapter 2.3 includes 80 full step-by-step 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 step-by-step 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.
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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.
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Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
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Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
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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
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Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
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Gauss-Jordan method.
Invert A by row operations on [A I] to reach [I A-I].
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Incidence matrix of a directed graph.
The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .
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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.
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Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
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Left inverse A+.
If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.
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Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
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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.
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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).
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Reflection matrix (Householder) Q = I -2uuT.
Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.
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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.
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Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
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Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.
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Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).
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Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.
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Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).