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1. Consider the functions f : A → B and g : B → C. a. If f and g are both injective, show that so is go f. b. Suppose that gof is injective. Are f and g necessarily injective? Prove or provide a counterexample. c. If f and g are both surjective, show that so is g o f. d. Suppose that gof is surjective. Are f and g necessarily surjective? Prove or provide a counterex- ample.

1. Consider the functions f : A → B and g : B → C. a. If f and g are both injective, show that so is go f. b. Suppose that gof is injective. Are f and g necessarily injective? Prove or provide a counterexample. c. If f and g are both surjective, show that so is g o f. d. Suppose that gof is surjective. Are f and g necessarily surjective? Prove or provide a counterex- ample.

Advanced Engineering Mathematics
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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Answer a and b

1. Consider the functions f : A → B and g : B → C. a. If f and g are both injective, show that so is go f. b. Suppose that gof is injective. Are f and g necessarily injective? Prove or provide a counterexample. c. If f and g are both surjective, show that so is g o f. d. Suppose that gof is surjective. Are f and g necessarily surjective? Prove or provide a counterex- ample.
Transcribed Image Text:1. Consider the functions f : A → B and g : B → C. a. If f and g are both injective, show that so is go f. b. Suppose that gof is injective. Are f and g necessarily injective? Prove or provide a counterexample. c. If f and g are both surjective, show that so is g o f. d. Suppose that gof is surjective. Are f and g necessarily surjective? Prove or provide a counterex- ample.
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