Function composition | Mathematics | Page 1

jaunty wyvernBOT Dec 4, 2024, 10:57 AM

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full crater Dec 4, 2024, 11:02 AM

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doesn't have to be in general

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how are your f and g defined?

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@digital edge

digital edge Dec 4, 2024, 11:15 AM

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It is about a characterization of injectivity.

full crater Dec 4, 2024, 11:22 AM

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but you asked why is fg = ..

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what do you assume?

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give the entire problem statement

digital edge Dec 4, 2024, 11:36 AM

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I should prove that this statement is equivalent to the statement that f is injective, however, I don’t understand the statement. I thought that the input of a function is supposed to be the domain and the output the codomain. So that g(b)=a, and therefore, f(a)=b, which would be ∆B, but it’s not.

full crater Dec 4, 2024, 11:39 AM

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what is Delta A?

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identity?

digital edge Dec 4, 2024, 11:40 AM

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Yes

full crater Dec 4, 2024, 11:43 AM

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that statement is all sorts of whack then

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We have a function f:A->B

f is injective if and only if there exists g: B->A such that gf = 1_A

digital edge Dec 4, 2024, 11:46 AM

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What does 1_A mean?

full crater Dec 4, 2024, 11:46 AM

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Delta A

digital edge Dec 4, 2024, 11:46 AM

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Ahh

full crater Dec 4, 2024, 11:46 AM

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1_A(a) = a for all a in A

digital edge Dec 4, 2024, 11:47 AM

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For surjective it says g o f = 1_B

full crater Dec 4, 2024, 11:48 AM

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in what order do you wrote compositions?

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$$ (g\circ f)(a) = g(f(a)) $$

quasi vineBOT Dec 4, 2024, 11:48 AM

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aL

full crater Dec 4, 2024, 11:48 AM

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do you define like this?

digital edge Dec 4, 2024, 11:48 AM

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Yes, that is how i would define it

full crater Dec 4, 2024, 11:49 AM

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then gf = 1_B is nonsense

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unless A=B

digital edge Dec 4, 2024, 11:49 AM

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I think i need to ask my prof

full crater Dec 4, 2024, 11:50 AM

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f:A->B is surjective iff there exists g:B->A such that fg = 1_B

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whenever you have composition gf = 1_A, then the first term is injective and last term is surjective

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so f is injective since that's what you apply first

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and g is surjective, becaus you apply it last

digital edge Dec 4, 2024, 11:51 AM

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I think so too

full crater Dec 4, 2024, 11:52 AM

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digital edge For surjective it says g o f = 1_B

this is nonsense already because B is not the domain of f

digital edge Dec 4, 2024, 11:58 AM

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I guess I have to skip the homework and maybe reach out to my professor because I am absolutely confused how that can be

full crater Dec 4, 2024, 11:58 AM

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maybe he denotes compositions in reverse order in his class

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regardless

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this is not a complete problem statement

digital edge Dec 4, 2024, 12:00 PM

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I have no clue

full crater Dec 4, 2024, 12:03 PM

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double check with classmates what the actual problem statement is

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i refuse to believe that's all what your professor told you

digital edge Dec 4, 2024, 12:04 PM

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Or maybe the denotation is really different

full crater Dec 4, 2024, 12:05 PM

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I'm not wasting time on speculating, find out what the actual problem is

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keksalute

digital edge Dec 4, 2024, 12:16 PM

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Thank you for your time, I appreciate your help. I’ll continue to look into it.

#Function composition