(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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Please do exercise 8.6.19 part abc. Please show step by step and explain.
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.
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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