5. (a) Suppose f: N Z is a one-to-one onto function.Prove that the function g: N x N N x Z definedby g(m, n) = (m, f (n)) is one-to-one and onto.%3D(b) Find a one-to-one correspondence between N x Nand N x Z.2 dt avm

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Answered: 5. (a) Suppose f: N→ Z is a one-to-one…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Suppose f and g are both functions.
4. Sketch the graph of an example function f on domain [0, 10] with jump discontinuities at x = 3 (righ continuous), x = 5 (left continuous), and x = 7 (neither left nor right continuous), where f(3) = 3, f (5) = 5, f (7) = 7.
This problem is about giving an example of a function from Z to Z that is onto, but not one-to-one. Select one of the following choices: (a) x + [x]-[x]. (c) 2 x/2 + 1. (b) [x/2 7. (d) [x].
(6a) how many different functions f : A → B can be defined if m=3 and n=5? (6b) if m=4 and n=6 then how many different one-to-one functions f : A →B can be defined? (6c) if m=5 and n=4 then how many different onto functions f : A → B can be defined?
24
2. Consider the function y(x) = 5x over the interval [0, 10]. (a) If the points A(0, y(0)), B(5, y(5)), and C(10, y(10)) are on the curve represented by the above function, find the slope of the lines (secants) AB, and AC, respectively. (b) If x represents time and y represents the displacement of a particle moving along a line, what is the physical meaning of your results to part (a)? (c) Now consider a point D(10 – h, y(10 – h)) on the curve represented by the given function, find the slope of the line DC. (d) As h → 0, what is the limiting value of your result to part (c)? (e) What is its physical meaning of your result to part (d) if again x represents time and y represents the displacement?
13. (9 points) Let D be the set of finite subsets of positive integers. Let S be the set of all positive integers greate than or equal to 2. Define a function T:S → D as follows: For each integer n ≥ 2, T(n) = the set of all even factors of n. a) Find T(10). b) Find T(17) c) Find T(m), where m is any odd positive integer.
2. Let f: R→R be defined by f(x) = 4x -x2. Let B be the range of f. (a) Find B. (b) Fins f-¹([0, ∞ )). (c) Find a set ASR such that fl, the restriction of f on A, is a one-to-one and onto function from A to B, and find (fl)

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5. (a) Suppose f: N→ Z is a one-to-one onto function. Prove that the function g: N × N → N x Z defined (m, f (n)) is one-to-one and onto. by g(m, n) = (b) Find a one-to-one correspondence between N x N and N x Z.