3.(a)(b)Let f: X→ Y and g: Y→ Z be functions. Prove the followingstatements.If f and g are both injective, then go f is injective.If f and g are both surjective, then go f is surjective.(c) If ƒ and g are both bijective, then go f is bijective.

We can prove the following parts .

Answered: 3. (a) Let f: X → Y and g: Y→ Z be…

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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From the statements below select the correct ones. □ A function f from A to B is one-to-one iff for all x, y = A, x = y implies that f(x) = f(y). A function from X to Y is injective iff every element from X gets mapped to only one element in Y. □ A function f from X to Y is injective iff for all a,b ≤ X, f(a) = f(b) implies that a = b. □ A function from A to B is injective iff different elements from A get mapped to different elements of B. O None of the above
(Discrete Math)
. 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
Let f : A → B and g : B → C be functions. Prove that if f and g are bijective, theng ◦f is bijective. Let f : A →B and g : B →C be functions.• Prove that if g ◦ f is surjective then g is surjective.• Prove that if g ◦ f is bijective, then f is injective and g is surjective.• Find functions f and g such that g ◦f is bijective but f is not surjective and g is notinjective. Hence the preceding statements are sharp.
DISCRETE MATHEMATICS:
Let S = {0,1}. Exhibit all functions S → S and classify them as injective, surjective, bijective, or none.

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3. (a) Let f: X → Y and g: Y→ Z be functions. Prove the following statements. If f and g are both injective, then go f is injective. (b) If f and g are both surjective, then go f is surjective. (c) If f and g are both bijective, then go f is bijective.