 7.2.1: The definition of onetoone is stated in two ways: x1, x2 X, if F(...
 7.2.2: Fill in each blank with the word most or least. a. A function F is ...
 7.2.3: When asked to state the definition of onetoone, a student replies...
 7.2.4: Let f : X Y be a function. True or false? A sufficient condition fo...
 7.2.5: All but two of the following statements are correct ways to express...
 7.2.6: Let X = {1, 5, 9} and Y = {3, 4, 7}. a. Define f : X Y by specifyin...
 7.2.7: Let X = {a, b, c, d} and Y = {e, f, g}. Define functions F and G by...
 7.2.8: Let X = {a, b, c} and Y = {w, x, y,z}. Define functions H and K by ...
 7.2.9: Let X = {1, 2, 3}, Y = {1, 2, 3, 4}, and Z = {1, 2}. a. Define a fu...
 7.2.10: a. Define f : Z Z by the rule f (n) = 2n, for all integers n. (i) I...
 7.2.11: a. Define g: Z Z by the rule g(n) = 4n 5, for all integers n. (i) I...
 7.2.12: a. Define F: Z Z by the rule F(n) = 2 3n, for all integers n. (i) I...
 7.2.13: a. Define H: R R by the rule H(x) = x 2, for all real numbers x. (i...
 7.2.14: Explain the mistake in the following proof. Theorem: The function f...
 7.2.15: In each of 1518 a function f is defined on a set of real numbers. D...
 7.2.16: In each of 1518 a function f is defined on a set of real numbers. D...
 7.2.17: In each of 1518 a function f is defined on a set of real numbers. D...
 7.2.18: In each of 1518 a function f is defined on a set of real numbers. D...
 7.2.19: Referring to Example 7.2.3, assume that records with the following ...
 7.2.20: Define Floor: R Z by the formula Floor(x) = x, for all real numbers...
 7.2.21: Let S be the set of all strings of 0s and 1s, and define l: S Znonn...
 7.2.22: Let S be the set of all strings of 0s and 1s, and define D: S Z as ...
 7.2.23: Define F: P({a, b, c}) Z as follows: For all A in P({a, b, c}), F(A...
 7.2.24: Let S be the set of all strings of as and bs, and define N: S Z by ...
 7.2.25: Let S be the set of all strings in as and bs, and define C: S S by ...
 7.2.26: Define S: Z+ Z+ by the rule: For all integers n, S(n) = the sum of ...
 7.2.27: Let D be the set of all finite subsets of positive integers, and de...
 7.2.28: Define G: R R R R as follows: G(x, y) = (2y, x) for all (x, y) R R....
 7.2.29: Define H: R R R R as follows: H(x, y) = (x + 1, 2 y) for all (x, y)...
 7.2.30: Define J : Q Q R by the rule J (r,s) = r + 2s for all (r,s) Q Q. a....
 7.2.31: Define F: Z+ Z+ Z+ and G: Z+ Z+ Z+ as follows: For all (n, m) Z+ Z+...
 7.2.32: a. Is log8 27 = log2 3? Why or why not? b. Is log16 9 = log4 3? Why...
 7.2.33: The properties of logarithm established in 3335 are used in Section...
 7.2.34: The properties of logarithm established in 3335 are used in Section...
 7.2.35: The properties of logarithm established in 3335 are used in Section...
 7.2.36: Exercises 36 and 37 use the following definition: If f : R R and g:...
 7.2.37: Exercises 36 and 37 use the following definition: If f : R R and g:...
 7.2.38: Exercises 38 and 39 use the following definition: If f : R R is a f...
 7.2.39: Exercises 38 and 39 use the following definition: If f : R R is a f...
 7.2.40: Suppose F: X Y is onetoone. a. Prove that for all subsets A X, F1...
 7.2.41: Suppose F:X Y is onto. Prove that for all subsets B Y, F(F1(B)) = B.
 7.2.42: Let X = {a, b, c, d, e} and Y = {s, t, u, v, w}. In each of 42 and ...
 7.2.43: Let X = {a, b, c, d, e} and Y = {s, t, u, v, w}. In each of 42 and ...
 7.2.44: In 4455 indicate which of the functions in the referenced exercise ...
 7.2.45: In 4455 indicate which of the functions in the referenced exercise ...
 7.2.46: In 4455 indicate which of the functions in the referenced exercise ...
 7.2.47: In 4455 indicate which of the functions in the referenced exercise ...
 7.2.48: In 4455 indicate which of the functions in the referenced exercise ...
 7.2.49: In 4455 indicate which of the functions in the referenced exercise ...
 7.2.50: In 4455 indicate which of the functions in the referenced exercise ...
 7.2.51: In 4455 indicate which of the functions in the referenced exercise ...
 7.2.52: In 4455 indicate which of the functions in the referenced exercise ...
 7.2.53: In 4455 indicate which of the functions in the referenced exercise ...
 7.2.54: In 4455 indicate which of the functions in the referenced exercise ...
 7.2.55: In 4455 indicate which of the functions in the referenced exercise ...
 7.2.56: In Example 7.2.8 a onetoone correspondence was defined from the p...
 7.2.57: Write a computer algorithm to check whether a function from one fin...
 7.2.58: Write a computer algorithm to check whether a function from one fin...
Solutions for Chapter 7.2: OnetoOne and Onto, Inverse Functions
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 7.2: OnetoOne and Onto, Inverse Functions
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: 4th. Discrete Mathematics with Applications was written by Sieva Kozinsky and is associated to the ISBN: 9780495391326. Since 58 problems in chapter 7.2: OnetoOne and Onto, Inverse Functions have been answered, more than 24462 students have viewed full stepbystep solutions from this chapter. Chapter 7.2: OnetoOne and Onto, Inverse Functions includes 58 full stepbystep solutions.

Column space C (A) =
space of all combinations of the columns of A.

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

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

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

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

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

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

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.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

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

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

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.

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.

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

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

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.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.