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# A question about composite functions watch

1. 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?
2. (Original post by Simplicity)
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?
just because f doesn't map to 2 doesn't mean tha g shouldn't map 2 to anything.

WHAT ARE YOU CRYING ABOUT?
3. The composition of functions f and g described on wikipedia is fine. 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).
4. What are you stuck up about? The function g takes elements from the set {1, 2, 3, 4} and maps them to the set {@, #, !!}.

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 #.

EDIT: ARE YOU HAPPY NOW?!?
5. take f(x)=|x| and g(x)=x
g maps -anything to -anything but f(g(x)) doesn't map anything to -anything.

Attached Images

6. amended my post accordingly.
7. (Original post by around)
amended my post accordingly.
8. (Original post by Totally Tom)
just because f doesn't map to 2 doesn't mean tha g shouldn't map 2 to anything.
Okay. I guess that isn't that big of a problem.

(Original post by Drederick Tatum)
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).
What if and

then .

Yeah, I guess I will think about it like that.

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.

(Original post by around)
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 #.
Doesn't c get mapped to #. As c goes to 3 then #.
9. (Original post by Simplicity)
...

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.
This.
10. Yeah, but that completely destroys my picture of what a function is. Sort of like classical physicist realising that gravity means a atom would implode as electron would collide with the atom if classical mechanics was true.
11. (Original post by Simplicity)

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.
12. (Original post by around)
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.
What was that about Totally Tom and you? weird.

How are you visualizing this? Because, this sorts of destroys the picture. I guess fg is really one thing. And that the only problem is that gf can't be possible.
13. (Original post by Simplicity)
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.
Spoiler:
Show

Anyway, everyone starts off with silly misconceptions like this. It's fine for a while, but you have to get out of the habit now, I'm afraid. Think of a function as a 'rule' that accepts certain inputs and gives you certain outputs. Doesn't matter what this rule is, as long as it's well defined.
14. Anyway, everyone starts off with silly misconceptions like this. It's fine for a while, but you have to get out of the habit now, I'm afraid. Think of a function as a 'rule' that accepts certain inputs and gives you certain outputs. Doesn't matter what this rule is, as long as it's well defined.
Maybe, I have to stop watching House. There must be a childish way to picture it. All my knowledge of set theory is based on people putting objects into bags and stuff. Although, I don't visualise a bag as that would be pointless more like if you have , I would say take Y out of bag then put Y and X back in or something along that line.

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.
P.P.S. I was thinking of trying to think more abstract. But, I sort of went against this as I thought that would be counter productive.
15. (Original post by Simplicity)
All my knowledge of set theory is based on people putting objects into bags and 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...

(Original post by Simplicity)
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. 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.
16. (Original post by generalebriety)
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...
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. A better example, would be induction and the ladder analogy. Maybe, complex numbers in the form of polar-cordinates is another good example.

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.

(Original post by generalebriety)
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.
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. Set theory might not correspond to anything, but I been reading some things about set theory that suggest its not the foundation of mathematics. Certainly, pure mathematics has application to real world, so I don't see why you can't use applications of the pure maths to think about the pure maths part.

I read for example that Group theory is really important and could be understood through Physics.
17. (Original post by Simplicity)
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.
http://en.wikipedia.org/wiki/Surjective_function: look at the gallery (especially the 4th image).

also
(Original post by Simplicity)
I read for example that Group theory is really important and could be understood through Physics.
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.
18. (Original post by Simplicity)
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.
But you don't seem to understand set theory, so how can you possibly make this judgement?

I think it goes back to comments by Einstein.

~snip~

I read for example that Group theory is really important and could be understood through Physics.
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.
19. (Original post by around)
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.
Thats what I meant.

(Original post by DFranklin)
But you don't seem to understand set theory, so how can you possibly make this judgement?
I know naive set theory. I have a book on set theory and it doesn't look that hard.

(Original post by DFranklin)
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.
I do know that. But, still I don't see why you have to purge metaphors out of thought to become a mathematician. Yeah, I know that was ancedotal evidence.
20. (Original post by Simplicity)
Going back to the example is gf a surjection? as f isn't a surjection.
gf: {a, b, c, d} --> {@, #, !!} is defined by
gf(a) = @,
gf(b) = @,
gf(c) = #,
gf(d) = !!.

Does this look surjective?

(Original post by Simplicity)
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.
Einstein was a physicist. He's rumoured to have been a bit **** at maths, actually.

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