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(a) Suppose f: A → B and g: B → C. Show that if f and g are one-to-one, then gof is one-to-one. (b) Suppose f: A → B and g: B → C. Show that if g o f is one-to-one, then f is one-to-one. (c) Suppose f: A→ B and g: B → C. Show that if g o f is onto, then g is onto.

(a) Suppose f: A → B and g: B → C. Show that if f and g are one-to-one, then gof is one-to-one. (b) Suppose f: A → B and g: B → C. Show that if g o f is one-to-one, then f is one-to-one. (c) Suppose f: A→ B and g: B → C. Show that if g o f is onto, then g is onto.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.3: Properties Of Composite Mappings (optional)
Problem 2TFE: Label each of the following statements as either true or false. The composition of two bijections is...
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Please do exercise 8.6.19 part abc. Please show step by step and explain.

 

Exercise 8.6.19. Suppose f: A B and g: B → C. Use the properties from Exercise 8.6.18 to prove the following: (a) Show that if ƒ and g are bijections, then gof is a bijection. (b) Show that if f and gof are bijections, then g is a bijection. (c) Show that if g and gof are bijections, then f is a bijection.
Transcribed Image Text:Exercise 8.6.19. Suppose f: A B and g: B → C. Use the properties from Exercise 8.6.18 to prove the following: (a) Show that if ƒ and g are bijections, then gof is a bijection. (b) Show that if f and gof are bijections, then g is a bijection. (c) Show that if g and gof are bijections, then f is a bijection.
Exercise 8.6.18. (a) Suppose f: A → B and g: B → C. Show that if f and g are one-to-one, then gof is one-to-one. (b) Suppose f: A → B and g: B → C. Show that if go f is one-to-one, then f is one-to-one. (c) Suppose f: A → B and g: B → C. Show that if go f is onto, then g is onto. (d) Give an example of functions f: A → B and g: B → C, such that gof is onto, but ƒ is not onto. (e) Suppose f: A → B and g: B → C. Show that if g of is onto, and g is one-to-one, then ƒ is onto. (f) Suppose f: A → B and g: B → C. Show that if f is onto and go f is 1-1, then g is 1-1. (g) Define f: [0, 0) → R by f(x) = x. Find a function g: R → R such that gof is one-to-one, but g is not one-to-one. (h) Suppose f and g are functions from A to A. If f(a) = a for every a € A, then what are f og and go f?
Transcribed Image Text:Exercise 8.6.18. (a) Suppose f: A → B and g: B → C. Show that if f and g are one-to-one, then gof is one-to-one. (b) Suppose f: A → B and g: B → C. Show that if go f is one-to-one, then f is one-to-one. (c) Suppose f: A → B and g: B → C. Show that if go f is onto, then g is onto. (d) Give an example of functions f: A → B and g: B → C, such that gof is onto, but ƒ is not onto. (e) Suppose f: A → B and g: B → C. Show that if g of is onto, and g is one-to-one, then ƒ is onto. (f) Suppose f: A → B and g: B → C. Show that if f is onto and go f is 1-1, then g is 1-1. (g) Define f: [0, 0) → R by f(x) = x. Find a function g: R → R such that gof is one-to-one, but g is not one-to-one. (h) Suppose f and g are functions from A to A. If f(a) = a for every a € A, then what are f og and go f?
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