how do i gain intuition regarding if a function is injective or surjective before proving it | Mathematics | Page 1

steel thunder Dec 12, 2024, 4:45 AM

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when doing a proof, it is helpful to know if a function is injective or surjective b4hand, how do i do that

raw ridgeBOT Dec 12, 2024, 4:45 AM

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steel thunder when doing a proof, it is helpful to know if a function is injective or surjecti...

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bleak crag Dec 12, 2024, 4:58 AM

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Depends on how complicated the function is

steel thunder Dec 12, 2024, 4:59 AM

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wdym

magic hound Dec 12, 2024, 4:59 AM

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what sort of function?

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continuous? discrete?

steel thunder Dec 12, 2024, 5:00 AM

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i think discrete

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like with a domain and codomain

magic hound Dec 12, 2024, 5:00 AM

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give the function

steel thunder Dec 12, 2024, 5:00 AM

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i'm asking a general question

magic hound Dec 12, 2024, 5:00 AM

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by definition, then

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as always

steel thunder Dec 12, 2024, 5:01 AM

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using this definition of a func

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what

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r u sure

magic hound Dec 12, 2024, 5:05 AM

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eh no nvm, that's something different

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that's a function

steel thunder Dec 12, 2024, 5:05 AM

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yeah so what im saying is

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if u have intuition regarding if a function is injective/surjective, then it's easier to prove if it is

magic hound Dec 12, 2024, 5:06 AM

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there is no general "algorithm" that will determine injectivity or surjectivity

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and there is no finite list of "intuitions" to know whether it's injective or surjective

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hence the question: what function, exactly?

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you can always try by definition

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$$ \forall x,y (f(x) = f(y) \Rightarrow x=y) $$

warm mistBOT Dec 12, 2024, 5:08 AM

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aL

magic hound Dec 12, 2024, 5:08 AM

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if you have a continuous function then it's enough to check it's strictly monotone

steel thunder Dec 12, 2024, 5:12 AM

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Bc it depends on the domain and codomain right

magic hound Dec 12, 2024, 5:19 AM

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i don't know what it means for "it to depend on the comain or codomain"

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but you are right, that changing domain can have an effect on injectivity

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$$ [0,\infty) \to\mathbb R,\quad x\mapsto x^2 $$

warm mistBOT Dec 12, 2024, 5:20 AM

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aL

magic hound Dec 12, 2024, 5:20 AM

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this is injective

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whereas

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$$ \mathbb R\to\mathbb R,\quad x\mapsto x^2 $$

warm mistBOT Dec 12, 2024, 5:21 AM

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aL

magic hound Dec 12, 2024, 5:21 AM

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is not injective

steel thunder Dec 12, 2024, 5:22 AM

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Similarly, Changing the codomain has an effect on surjectivity

magic hound Dec 12, 2024, 5:22 AM

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correct

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but you can't change these willy nilly

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you must always have a well defined function to even ask whether it's inj/surj

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i.e this

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steel thunder Dec 12, 2024, 5:24 AM

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Can u give an example

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Of where changing the codomain would cause the function to no longer be one

magic hound Dec 12, 2024, 5:26 AM

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$$ \mathbb R \to [1,\infty),\quad x\mapsto x^2 $$

warm mistBOT Dec 12, 2024, 5:26 AM

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aL

magic hound Dec 12, 2024, 5:26 AM

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is ill defined

steel thunder Dec 12, 2024, 5:27 AM

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Wdym Ill defined

magic hound Dec 12, 2024, 5:27 AM

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doesn't satisfy this requirement

steel thunder Dec 12, 2024, 5:27 AM

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Oh

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Wait what does the exclamation mean

magic hound Dec 12, 2024, 5:28 AM

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exists exactly one y

steel thunder Dec 12, 2024, 5:28 AM

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Ok

magic hound Dec 12, 2024, 5:28 AM

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if you put a vertical line through the graph, then it must have exactly one intersection point

steel thunder Dec 12, 2024, 5:28 AM

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And why does what u posted violate that

magic hound Dec 12, 2024, 5:28 AM

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as you're probably taught in high school

steel thunder Dec 12, 2024, 5:28 AM

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Yeh

magic hound Dec 12, 2024, 5:29 AM

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steel thunder And why does what u posted violate that

you tell me

steel thunder Dec 12, 2024, 5:30 AM

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What what is the atrow

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Arrow

magic hound Dec 12, 2024, 5:30 AM

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it's just notation

steel thunder Dec 12, 2024, 5:31 AM

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No I meant

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Oh

magic hound Dec 12, 2024, 5:31 AM

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$$ f:\mathbb R \to [1,\infty),\quad f(x) = x^2 $$

warm mistBOT Dec 12, 2024, 5:31 AM

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aL

steel thunder Dec 12, 2024, 5:31 AM

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Yeah

magic hound Dec 12, 2024, 5:31 AM

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same thing

steel thunder Dec 12, 2024, 5:31 AM

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Got it

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Idrk man

magic hound Dec 12, 2024, 5:32 AM

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you are missing something in the codomain

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either way, gtg work, think about it

steel thunder Dec 12, 2024, 5:34 AM

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Oh

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Because x = 0 doesn’t map to anything

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Duh

burnt hornetBOT Dec 13, 2024, 6:00 AM

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@steel thunder

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#how do i gain intuition regarding if a function is injective or surjective before proving it