 7.1.1: Let X = {1, 3, 5} and Y = {s, t, u, v}. Define f : X Y by the follo...
 7.1.2: Let X = {1, 3, 5} and Y = {a, b, c, d}. Define g: X Y by the follow...
 7.1.3: Indicate whether the statements in parts (a)(d) are true or false. ...
 7.1.4: a. Find all functions from X = {a, b} to Y = {u, v}. b. Find all fu...
 7.1.5: Let IZ be the identity function defined on the set of all integers,...
 7.1.6: Find functions defined on the set of nonnegative integers that defi...
 7.1.7: Let A = {1, 2, 3, 4, 5} and define a function F: P(A)Z as follows: ...
 7.1.8: Let J5 = {0, 1, 2, 3, 4}, and define a function F: J5 J5 as follows...
 7.1.9: Define a function S : Z+ Z+ as follows: For each positive integer n...
 7.1.10: Let D be the set of all finite subsets of positive integers. Define...
 7.1.11: Define F : Z ZZ Z as follows: For all ordered pairs (a, b) of integ...
 7.1.12: Define G : J5 J5 J5 J5 as follows: For all (a, b) J5 J5, G(a, b) = ...
 7.1.13: Define G : J5 J5 J5 J5 as follows: For all (a, b) J5 J5, G(a, b) = ...
 7.1.14: Let J5 = {0, 1, 2, 3, 4}, and define functions h : J5 J5 and k : J5...
 7.1.15: Let F and G be functions from the set of all real numbers to itself...
 7.1.16: Let F and G be functions from the set of all real numbers to itself...
 7.1.17: Use the definition of logarithm to fill in the blanks below. a. log...
 7.1.18: Find exact values for each of the following quantities. Do not use ...
 7.1.19: Use the definition of logarithm to prove that for any positive real...
 7.1.20: Use the definition of logarithm to prove that for any positive real...
 7.1.21: If b is any positive real number with b = 1 and x is any real numbe...
 7.1.22: Use the unique factorization for the integers theorem (Section 4.3)...
 7.1.23: If b and y are positive real numbers such that logb y = 3, what is ...
 7.1.24: If b and y are positive real numbers such that logb y = 2, what is ...
 7.1.25: Let A = {2, 3, 5} and B = {x, y}. Let p1 and p2 be the projections ...
 7.1.26: Observe that mod and div can be defined as functions from Znonneg Z...
 7.1.27: Let S be the set of all strings of as and bs. a. Define f : S Z as ...
 7.1.28: Consider the coding and decoding functions E and D defined in Examp...
 7.1.29: Consider the Hamming distance function defined in Example 7.1.10. a...
 7.1.30: Draw arrow diagrams for the Boolean functions defined by the follow...
 7.1.31: Fill in the following table to show the values of all possible two...
 7.1.32: Consider the threeplace Boolean function f defined by the followin...
 7.1.33: Student A tries to define a function g: Q Z by the rule g m n = m n...
 7.1.34: Student C tries to define a function h: Q Q by the rule h m n = m2 ...
 7.1.35: Let J5 = {0, 1, 2, 3, 4}. Then J5 {0}={1, 2, 3, 4}. Student A tries...
 7.1.36: Let J4 = {0, 1, 2, 3}. Then J4 {0}={1, 2, 3}. Student C tries to de...
 7.1.37: On certain computers the integer data type goes from 2, 147, 483, 6...
 7.1.38: Let X = {a, b, c} and Y = {r,s, t, u, v, w}. Define f : X Y as foll...
 7.1.39: Let X = {1, 2, 3, 4} and Y = {a, b, c, d, e}. Define g: X Y as foll...
 7.1.40: Let X and Y be sets, let A and B be any subsets of X, and let F be ...
 7.1.41: In 4149 let X and Y be sets, let A and B be any subsets of X, and l...
 7.1.42: In 4149 let X and Y be sets, let A and B be any subsets of X, and l...
 7.1.43: In 4149 let X and Y be sets, let A and B be any subsets of X, and l...
 7.1.44: In 4149 let X and Y be sets, let A and B be any subsets of X, and l...
 7.1.45: In 4149 let X and Y be sets, let A and B be any subsets of X, and l...
 7.1.46: In 4149 let X and Y be sets, let A and B be any subsets of X, and l...
 7.1.47: In 4149 let X and Y be sets, let A and B be any subsets of X, and l...
 7.1.48: In 4149 let X and Y be sets, let A and B be any subsets of X, and l...
 7.1.49: In 4149 let X and Y be sets, let A and B be any subsets of X, and l...
 7.1.50: Given a set S and a subset A, the characteristic function of A, den...
 7.1.51: Each of exercises 5153 refers to the Euler phi function, denoted , ...
 7.1.52: Each of exercises 5153 refers to the Euler phi function, denoted , ...
 7.1.53: Each of exercises 5153 refers to the Euler phi function, denoted , ...
Solutions for Chapter 7.1: Functions Defined on General Sets
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 7.1: Functions Defined on General Sets
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4. Since 53 problems in chapter 7.1: Functions Defined on General Sets have been answered, more than 49058 students have viewed full stepbystep solutions from this chapter. Discrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326. Chapter 7.1: Functions Defined on General Sets includes 53 full stepbystep solutions.

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

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + 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.

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

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

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

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(DÂ» O.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

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.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

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.

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

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.