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I was given in homework to formalize some statements regarding functions, but after a long while scanning wikipedia, I wasn't able to find how to start doing so, any help would be appreciated, will just give an example, I'll fill out the rest myself :)

example: There exists groups A, B so that every function from A to B is injective.

There exists f: A -> B so that f is injective but not surjective.

Hope it's understood, because I'm completely clueless...

EDIT: perhaps formal language wasn't an appropriate tag, but what I'm referring to is the "translation" of plain english to logical mathametical statements.

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closed as not a real question by Did, Fabian, Austin Mohr, Asaf Karagila, J. M. Jan 13 '12 at 3:17

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

    
What do you mean by formalization? What is the formal language you are working on? You used the tag "elementary set theory". Does that mean you need to explain how to formalize these notions within set theory? If that is the case, do you know how to formalize ordered pairs? functions? algebraic structures? –  Andres Caicedo Jan 10 '12 at 22:56
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Why the down vote? –  Matt N. Jan 10 '12 at 22:57
    
Dear Sagi: Can you include more information in your question? If English is a problem try to post the additional information in your native language. Someone on SE will most likely be able to translate it for you! Cheers –  Matt N. Jan 10 '12 at 22:59

1 Answer 1

Kind of this?

It's bijection in my vision:

$f:A\rightarrow B \hspace{5mm} (\forall x_1, x_2 \in A \hspace{5mm} x_1\neq x_2 \Rightarrow f(x_1)\neq f(x_2)) \hspace{5mm} \text{&} \hspace{5mm} (\forall y\in B \hspace{5mm}\exists x\in A : f(x)=y)$

Yeah, without forall should be better..

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yup, like this, however i'm not sure that's the correct formalization. –  Itai Sagi Jan 10 '12 at 23:01
    
Not sure about the "for all f" quantifier, but what follows is essentially "f is injective & f is surjective –  The Chaz 2.0 Jan 10 '12 at 23:14

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