Can a Generalized Proof Show Containment Without Counter Example?

[SqrachMasda]

i don't know if generalize is the correct term but

f:A->B
C,C1,C2 are subsets of A and D,D1,D2 are subsets of B
(most of that is not needed for this part)

C (contained in) f-1[f(C)] (f-1, is f inverse. i got to learn symbols)

okay, my teacher always tells us we can use examples to help us understand it but we can't use examples to prove something. however he then proceeded to prove it with an example. which made it far too simple especially since it was the same one i chose and then chose not to use

I know if it's 1-1(injective) then it's going to be equal and not just contained in one directions
(i assume to be a function then it must be surjective, but I'm still not sure if that's correct to say)
so something like f: X^2 for some X in C would work
i had something like C={-2,-1...3} so the inverse funtion would have {-3} and C
but i spent a long time and a lot of space because I was trying to make the proof a more general statement

So, is it possible to give a more general proof to show it is contained in and would imply that it's not equal without actually having to counter example it is not equal?


I was definitely thrown off since he goes out of his way to say not to use examples and then summed them up with simple examples

 

[SqrachMasda]

hey, I'm not angry. wrong face

 

[matt grime]

It is very hard to decipher what it is your asking. In fact you don't actually ever ask a question.

I tihnk that you're asking:

Suppose that f is a function from A to B and that C is a subset of A. Show that C is a subset of f^-1(f(C)).

Now, this is trivial from the definition of f^-1. Recall that f^-1(D) is the set of x in A such that f(x) is in D. So the result is a clear and simple consequence of the definition.

I think that the second thing you're asking is to show that in general f^-1(f(C)) is not equal to C. But it suffices to provide one example to demonstrate this. The statement you want to contradict is a 'for all' statement, so a single counter example will contradict it.

 

[SqrachMasda]

thanks for kicking me in the face on the way in

okay, not the best reviewed post
it's obviously a true statement
i thought there would be more to proving it
but i see it now

i'm new to this
sorry I'm not at your level of god like mastery of the subject
99% of the answers always involve some level of arrogance
it's annoying

 

[matt grime]

I'm sorry you feel unduly hard done by, but you should look at your post objectively. Here is the opening part:

i don't know if generalize is the correct term but

f:A->B
C,C1,C2 are subsets of A and D,D1,D2 are subsets of B
(most of that is not needed for this part)

C (contained in) f-1[f(C)] (f-1, is f inverse. i got to learn symbols)

okay, my teacher always tells us we can use examples to help us understand it but we can't use examples to prove something. however he then proceeded to prove it with an example.

He then proceeded to prove what? You've not written out a theorem, lemma, proposition, or anything that implies a proof is what you are required to show.

Why put in C1,C2 etc?

If people have to actually first work out what it is you are asking before even starting on the solution, then you may well end up getting little help. Write clearly, preferably in sentences with punctuation, and you will find people a lot more willing to offer the answer.

Your teacher did not prove that for all f,C etc, that C is a subset of f^-1(f(C)) with an example. He will have proved that properly, and then shown by example that the containment may be strict.