Hi, this is my first post here so please tell me if there is something wrong about it...
I just started to read "Notes on set theory" by Yiannis Moschovakis which is available online through my university library and now I am very unsure even about the first few introductory exercise. I would really appreciate some verification of my way to think here, especially since I am not very good in math and haven't read any thorough courses in logic et.c. before. Because this is just an online book which I picked up and started reading, there are no lecturers or such to ask either, hence I turn to mathhelpforum =)
Also, sorry about this wall of text! But I guess I should post my entire attempt...
...in the end of the post I try to be a bit more specific!
I realize that I may be way off with this attempt and there may very well be MAJOR misconceptions and errors. I appreciate all input, big thx in advance!
An example of a problem is :"Show that for every injection f : X Y, and all X
f[A B] = f[A] f[B].
Also show that this is identity does not always hold if f is not an injection."
First attempt at solution:
____________________start of first attempt________________________________
if x f[A B], then there is some y A B such that f(y)=x.
y (A B) y A y B
Hence, x f[A B] y A y B
If x f[A] f[B], then there is some y´ A y´ B such that f(y´) = x.
That is, x f[A] f[B] y´ A y´ B
Since f is an injection (stated in the problem formulation); x=f(y)=f(y´) y´= y
Using this to evolve the result from above:
x f[A] f[B] y´ A y´ B y A y B x f[A B]
f[A B] = f[A] f[B]
If f isn't an injection, then x=f(y)=f(y´) does not imply y=y´.
________________________end of first attempt_____________________________
Then I started to think about an example wof when the relation does not hold and I realized that my solution might not be very good.
Hence, I tried to adjust my first attempt:
____________________start of augmentation___________________________
If x f[A] f[B], then there is some y´ A such that f(y´) = x and y´´ B such that f(y´´)=x.
Since f is an injection, x=f(y´)=f(y´´) implies y´=y´´.
y´ A y´ B
___________________________end of augmentation_____________________
Would this augmentation make the solution correct?
Then if y´ is not equal to y´´ , I think it would be quite easy to make a specific example of when the identity does not hold for a non-injective fcn. (for instance say f(x)=x^2 and y´=-y´´, right?)
What I´m concerned about is probably mainly this:
x f[A] f[B] y´ A y´ B yA yB
by the argument that x=f(y)=f(y´) implies y=y´for an injection.
x f[A B] yA yB
x f[A] f[B] yA yB.
is this even correct?? necessary?
Big thx in advance for any input! // HeadmasterEel