A question about composite functions

http://en.wikipedia.org/wiki/Function_composition

Basically, in g on the diagram on wikipedia 2 is mapped to #. But, what the hell?

Nothing gets mapped from X to 2. But, then something is mapped from 2 to #. Surely, this most break some mathematical rule or something, like dividing by zero.

Because it doesn't make sense in the ball in box analogy. Since a ball appears at 2 that is put in box #. But, nobody puts a ball in 2 because the balls at X isn't put in the box in 2.

This is totally destroying the ball in box analogy.

Surely, its undefined at 2 so shouldn't map to #. This is ball.

P.S. Also, isn't it a problem if two balls land in the same box? does that mean one ball is left behind.
P.P.S. All the three textbooks don't have this problem or don't explain what it means.
P.P.P.S. Does this mean I need to drop the ball in box analogy?

amended my post accordingly.

just because f doesn't map to 2 doesn't mean tha g shouldn't map 2 to anything.

And I think it's best to work with the definitions of functions, function composition etc. rather than this ball in box analogy (I don't know what this is by the way).

It doesn't matter that f doesn't map anything in the set {a, b, c, d} to 2, all this means is that fg won't map anything in the set {a, b, c, d} to #.

...

Ball in box analogy is this say f(x)=y, defined by the above function then say we have an x in X. x is like a ball. Say if f(x)=y in y in Y. y is like a box. So f(x)=y, is like taking a ball and putting it in a box i.e. x is the ball y is the box. This is used to explain why x can only be mapped to one y, as you only have one ball so you can't put it into two different boxes.

I guess it doesn't really make sense in composite functions.

Doesn't c get mapped to #. As c goes to 3 then #.

I just got shown who is the boss. By Simplicity.

i'm sure i had a point when i made my post, but you've made me reconsider. i think what i was trying to say is that nothing in {a, b, c, d} gets mapped to # by fg via 2.

All my knowledge of set theory is based on people putting objects into bags and stuff.

P.S. I still don't know why you have to get rid off picture and stuff. Because, wouldn't that make you a robot? Surely, if you truly understand something you can describe it with everyday reality instead of abstract stuff.

No offence, but this might be why you're finding it so difficult. Set theory is a third year undergraduate course at Cambridge. If it was as simple as putting objects into bags...

No. Why should pure maths (and set theory, of all things) correspond to anything tangible? If anything, I'd say this kind of oversimplification was an indication that you didn't understand it.

Going back to the example is gf a surjection? as f isn't a surjection. But cleary every element of X is mapped to an element of Z through Y. I guess without the analogy I would have found this alot easier.

I read for example that Group theory is really important and could be understood through Physics.

Okay, that was a bad example. I wondering why does set theory come really late? As I'm pretty sure someone with A level mathematics could handle the computational side of it.

Group theory is one of the easier to represent topics of pure maths: there are plenty of groups (like D_n) which can be visualised very simply (ie the group of symmetries of a regular polygon with n sides). No need to bring Physics into it.

But you don't seem to understand set theory, so how can you possibly make this judgement?

It's important to realise that reading books about the lives of physicists and mathematicians doesn't actually qualify you as a physicist or mathematician.

Going back to the example is gf a surjection? as f isn't a surjection.

I think it goes back to comments by Einstein. Something about if you understood something you could be able to make it so simple that bar maid could understand. I guess a crappy justification.