 7.2.32E: a. Is log8 27 = log2 3? Why or why not?b. Is log16 9 = log4 3? Why ...
 7.2.1E: The definition of onetoone is stated in two ways:?x1, x2 ? X, if ...
 7.2.2E: Fill in each blank with the word most or least.a. A function F is o...
 7.2.3E: When asked to state the definition of onetoone, a student replies...
 7.2.4E: Let f: X ? Y be a function. True or false? A sufficient condition f...
 7.2.5E: All but two of the following statements are correct ways to express...
 7.2.6E: Let X = {1, 5, 9} and Y = {3, 4, 7}.a. Define f: X ? Y by specifyin...
 7.2.7E: Let X = {a, b, c, d} and Y = {e, f, g}. Define functions F and G by...
 7.2.8E: Let X = {a, b, c} and Y = {w, x, y, z}. Define functions H and K by...
 7.2.9E: Let X = {1, 2, 3}, Y = {1, 2, 3, 4}, and Z = {1, 2}.a. Define a fun...
 7.2.10E: a. Define f: Z ? Z by the rule f(n)= 2n, for all integers n.(i) Is ...
 7.2.11E: a. Define g: Z ? Z by the rule g(n)= 4n ? 5, for all integers n.(i)...
 7.2.12E: a. Define F: Z ? Z by the rule F(n)= 2 ? 3n, for all integers n.(i)...
 7.2.13E: a. Define H: R ? R by the rule H(x)= x2, for all real numbers x.(i)...
 7.2.14E: Explain the mistake in the following “proof.”Theorem: The function ...
 7.2.15E: In a function f is defined on a set of real numbers. Determine whet...
 7.2.16E: In a function f is defined on a set of real numbers. Determine whet...
 7.2.17E: In a function f is defined on a set of real numbers. Determine whet...
 7.2.18E: In a function f is defined on a set of real numbers. Determine whet...
 7.2.19E: Referring to Example 7.2.3, assume that records with the following ...
 7.2.20E: Define Floor: R ? Z by the formula Floor for all real numbers x.a. ...
 7.2.21E: Let S be the set of all strings of 0’s and 1’s, and define l: S ? Z...
 7.2.22E: Let S be the set of all strings of 0’s and 1’s, and define D: S ? Z...
 7.2.23E: Define as follows: For all A in F(A)= the number of elements in A.a...
 7.2.24E: Let S be the set of all strings of a’s and b’s, and define N: S ? Z...
 7.2.25E: Let S be the set of all strings in a’s and b’s, and define C: S ? S...
 7.2.26E: Define S: Z+ ? Z+ by the rule: For all integers n, S(n) = the sum o...
 7.2.27E: Let D be the set of all finite subsets of positive integers, and de...
 7.2.28E: Define G: R × R ? R × R as follows:G(x, y) = (2y, ?x) for all (x, y...
 7.2.29E: Define H: R × R ? R × R as follows:H(x, y) = (x + 1, 2 ? y) for all...
 7.2.30E: Define J: Q × Q ? R by the rule J(r, s)= r + for all (r, s)? Q × Q....
 7.2.31E: Define F: Z+ × Z+ ? Z+ and G: Z+ × Z+ ? Z+ as follows: For all (n, ...
 7.2.33E: Prove that for all positive real numbers b, x, and y with b ? 1,
 7.2.34E: Prove that for all positive real numbers b, x, and y with b ? 1,log...
 7.2.35E: Prove that for all real numbers a, b, and x with b and x positive a...
 7.2.36E: Exercise use the following definition: If f: R ? R and g: R ? R are...
 7.2.37E: Exercise use the following definition: If f: R ? R and g: R ? R are...
 7.2.38E: Exercise use the following definition: If f: R ? R is a function an...
 7.2.39E: Exercise use the following definition: If f: R ? R is a function an...
 7.2.40E: Suppose F: X ? Y is onetoone.a. Prove that for all subsets A ? X,...
 7.2.41E: Suppose F:X ? Y is onto. Prove that for all subsets B ? Y, F(F1(B)...
 7.2.42E: Let X = {a, b, c, d, e} and Y = {s, t, u, v, w}. In a onetoone co...
 7.2.43E: Let X = {a, b, c, d, e} and Y = {s, t, u, v, w}. In a onetoone co...
 7.2.44E: Indicate which of the functions in the referenced exercise are one...
 7.2.45E: Indicate which of the functions in the referenced exercise are one...
 7.2.46E: Indicate which of the functions in the referenced exercise are one...
 7.2.47E: Indicate which of the functions in the referenced exercise are one...
 7.2.48E: In 44–55 indicate which of the functions in the referenced exercise...
 7.2.49E: In 44–55 indicate which of the functions in the referenced exercise...
 7.2.50E: In 44–55 indicate which of the functions in the referenced exercise...
 7.2.51E: In 44–55 indicate which of the functions in the referenced exercise...
 7.2.52E: Indicate which of the functions in the referenced exercise are one...
 7.2.53E: Indicate which of the functions in the referenced exercise are one...
 7.2.54E: Indicate which of the functions in the referenced exercise are one...
 7.2.55E: Indicate which of the functions in the referenced exercise are one...
 7.2.56E: In Example 7.2.8 a onetoone correspondence was defined from the p...
 7.2.57E: Write a computer algorithm to check whether a function from one fin...
 7.2.58E: Write a computer algorithm to check whether a function from one fin...
Solutions for Chapter 7.2: Discrete Mathematics with Applications 4th Edition
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 7.2
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Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

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.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

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

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.

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!).

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

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

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).

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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.

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

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.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Solvable system Ax = b.
The right side b is in the column space of A.

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

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.