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6. Let f : X →Y be a function. (a) Prove that if f is surjective and g1 : Y → Z and g2 : Y → Z are functions such that g1 o f = 92 o f, then g1 = 92. (b) Prove that if f is injective and h : W → X and h2 : W → X are functions such that foh1 = f o h2, then h1 = h,.

6. Let f : X →Y be a function. (a) Prove that if f is surjective and g1 : Y → Z and g2 : Y → Z are functions such that g1 o f = 92 o f, then g1 = 92. (b) Prove that if f is injective and h : W → X and h2 : W → X are functions such that foh1 = f o h2, then h1 = h,.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.5: The Kernel And Range Of A Linear Transformation
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Q6 Mathmatical reasoning please with full explanation and steps. thx. typed answers will be appreciated
1. (a) Let f : Z+ → R+ and g : Z4 → R+ be functions. We say that f E O(g) (“f is big-O of g", usually denoted f = 0(g) in computer science classes) if there exist constants c e R, and N E Z̟ such that f(n) < c· g(n) for all n > N. Write down a precise mathematical statement of what f ¢ O(g) means. (b) Let f : R –→ R be a function and let ro, L E R. We say that lim+ro f (x) = L if for all e > 0, there exists a 8 > 0 such that for all x E R, 0 < |x – xo| < 8 → \f(x) – L| < ɛ Write down a precise mathematical statement of what lim,-20 f(x) # L means. (It is typical to write e > 0 instead of ɛ E R in analysis classes). 2. Let X be a set, and let A C X. Define a function f : P(X) → P(X) by f(B) = AUB for all BE P(X). Show that im(f) = {C € P(X)|A C C}. 3. Let f : X →Y and g : Y → Z be functions. (a) Prove that if f and g are injective, then g o f is injective. (b) Prove that if g o f is injective, then f is injective. (c) Give an example of specific functions f and g for which g o f is injective, but g is not injective. 4. Let f : X –→Y and g : Y → Z be functions. (a) Prove that if f and g are surjective, then gof is surjective. (b) Prove that if gof is surjective, then g is surjective. (c) Give an example of specific functions f and g for which gof is surjective, but f is not surjective. 5. Suppose that f : X →Y and g : Y → Z are bijections. Prove that gof is a bijection and that (go f)-1 = f-l og¬1. 6. Let f : X →Y be a function. (a) Prove that if f is surjective and g1 : Y → Z and g2 : Y → Z are functions such that gi ° f = 92 o f, then g1 = 92. (b) Prove that if f is injective and h1 : W → X and h2 : W → X are functions such that foh1 = f o h2, then hi = h2. 7. Let X be a set with |X| = n E Z4. Prove that for any set Y, there exists a bijection f : X →Y. |Y| = n if and only if
Transcribed Image Text:1. (a) Let f : Z+ → R+ and g : Z4 → R+ be functions. We say that f E O(g) (“f is big-O of g", usually denoted f = 0(g) in computer science classes) if there exist constants c e R, and N E Z̟ such that f(n) < c· g(n) for all n > N. Write down a precise mathematical statement of what f ¢ O(g) means. (b) Let f : R –→ R be a function and let ro, L E R. We say that lim+ro f (x) = L if for all e > 0, there exists a 8 > 0 such that for all x E R, 0 < |x – xo| < 8 → \f(x) – L| < ɛ Write down a precise mathematical statement of what lim,-20 f(x) # L means. (It is typical to write e > 0 instead of ɛ E R in analysis classes). 2. Let X be a set, and let A C X. Define a function f : P(X) → P(X) by f(B) = AUB for all BE P(X). Show that im(f) = {C € P(X)|A C C}. 3. Let f : X →Y and g : Y → Z be functions. (a) Prove that if f and g are injective, then g o f is injective. (b) Prove that if g o f is injective, then f is injective. (c) Give an example of specific functions f and g for which g o f is injective, but g is not injective. 4. Let f : X –→Y and g : Y → Z be functions. (a) Prove that if f and g are surjective, then gof is surjective. (b) Prove that if gof is surjective, then g is surjective. (c) Give an example of specific functions f and g for which gof is surjective, but f is not surjective. 5. Suppose that f : X →Y and g : Y → Z are bijections. Prove that gof is a bijection and that (go f)-1 = f-l og¬1. 6. Let f : X →Y be a function. (a) Prove that if f is surjective and g1 : Y → Z and g2 : Y → Z are functions such that gi ° f = 92 o f, then g1 = 92. (b) Prove that if f is injective and h1 : W → X and h2 : W → X are functions such that foh1 = f o h2, then hi = h2. 7. Let X be a set with |X| = n E Z4. Prove that for any set Y, there exists a bijection f : X →Y. |Y| = n if and only if
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