"If g is onto, then go f is onto." For nonempty sets A, and B, let f: A → A and g: A → B be functions. Prove if the statements that are true, givecounterexamples to disprove those that are false.

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Answered: For nonempty sets A, and B, let f: A →A…

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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. 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.
We have shown that the composition of two surjective functions is surjective, i.e. that if ƒ : A → B and g : B → C are surjective, then go f : A → C is surjective. Now decide whether the following statements are true or false, and prove your claim. Remember, to prove that a statement is wrong, all you need to do is to give a counterexample. (i) If f : A → B and g: B → C are functions such that go f is surjective, then f is surjective. (ii) If f : A → B and g : B → C are functions such that go f is surjective, then g is surjective. Get started by drawing some pictures of sets with arrows. No need to submit your drawings.
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
Prove the following statements: (a) If f : A → B and g : B → C are one-to-one functions, then go f : A → C is one-to-one. (b) Use the definition to prove that for any sets A, B and C' if |A| < |C| and |C| < |B| then |A| < |B|.
For nonempty sets A, and B, let f: A →A and g: A → B be functions. Prove if the statements that are true, give counterexamples to disprove those that are false. "If f of is ono-to-one, then f is one-to-one."
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.
Is the set X of all functions f: Z+→ {0, 1} countable? Prove or disprove.
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.
Suppose f: R → R is a function from the set of real numbers to the same set with f(x) = x + 1. We write £ to represent fo f and fatl= fof. Is it true that £22o f = fo f?? Why? Is the set {g: R R | gof=fog} infinite? Why?

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For nonempty sets A, and B, let f: A →A and g: A → B be functions. Prove if the statements that are true, give counterexamples to disprove those that are false.