1*) Let X = {1,2, 3}, Y = {2,3, 4}, and Z = {1,2}.(a) Define a function f: Z → X that is one-to-one but not onto.(b) Define a function g: X → Z that is onto but not one-to-one.(c) State whether there exists a function h: X →Y that is one-to-one but not onto.

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Answered: 1*) Let X = {1,2, 3}, Y = {2,3, 4}, and…

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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Let A = {1,2,3, ...,8} and consider the function ƒ: f: P(A) → N given by ƒ (B) = |B|. f Prove or disprove the statement that f is one-to-one. (Inspired by Levin, 2016, p. 38).
A function g : Z → Z is defined as g(n) = |n| + 1. Show that this function is not injective and not surjective.
. For nonempty sets A, B and C, let f : A → B and g: B → C be functions. (a) Prove: If g of is one-to-one, then f is one-to-one. using each of the following proof techniques: direct proof, proof by contrapositive, proof by contradiction. (b) Disprove: If go f is one-to-one, then g is one-to-one.
Let f A → B be a function that is one-to-one and onto, and let C be any set such that CCA and f: C → B is still a function. True or False: f: C → B is still one-to-one. True False
(a) What is the number of onto functions f : {1,2, 3, 4, 5} → {1, 2, 3}? (b) What is the number of one to one functions f : {1,2, 3} → {1,2, 3, 4, 5}?
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Let A be a nonempty set and let f : A –→ B be a function. Prove that f is one-to-one if and only if there exists a function g : B → A so that gof = 14.
3. Let I(r) numbers such that if r # y then g(x) # g(y). Prove that there is a function f such that fog= I. = z be the compositive identity function. Suppose g is a function with domain of all real
Let f {1,2,..., 16} → {1,2,..., 7} be defined by the following tables: x f(x) 9 4 10 1 11 3 12 5 x f(x) 1 1 2 3 3 5 4 7 x f(x) 5 2 6 4 7 6 8 2 Define a function g {1,2,..., 7} g(f(x)) satisfies the following property: by a table and compute each of the sets h-¹({i}). x f(x) 2 1 13 14 15 3 16 1 {1,2,3,4} such that the composition h(x) |h¯¹({i})| = 4 for all i € {1,2,3,4}. Define g :
Let X {n ∈ N : 0 ≤ n ≤ 999} be the set of all numbers with three or fewer digits. Define the function f : X → N by f (abc) a + b + c, where a, b, and c are the digits of the number in X (write numbers less than 100 with leading 0’s to make them three digits). For example, f (253) 2 + 5 + 3 10. (a) Let A {n ∈ X : 113 ≤ n ≤ 122}. Find f (A). (b) Find f −1 ({1, 2}) (c) Find f −1 (3). (d) Find f −1 (28). (e) Is f injective? Explain. (f) Is f surjective? Explain.
1. Let A = {1,2,3}, B = B = {4,5,6}, The functions f : A → C and g: B → C are defined by f(1) = 7, f(2)= 9, f(3) = 7, Does there exist a function h: A → B such that go h = f? Does there exist a function k : B → A such that fo k = g? Justify your answers. {4,5,6}, C = {7, 8, 9, 10}. g(4) = 10, g(5) = 7, g(6) = 9.
Let f: A → B and g: B → C be functions. (a) Prove that if f : A → B is injective and g: B → C is injective, then go f is injective. (b) If f and go f are injective, is g always injective? Prove or provide a counterexample.

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1*) Let X = {1,2, 3}, Y = {2,3, 4}, and Z = {1,2}. (a) Define a function f: Z → X that is one-to-one but not onto. (b) Define a function g: X → Z that is onto but not one-to-one. (c) State whether there exists a function h: X →Y that is one-to-one but not onto.