Hello,
I saw somewhere that if we have a one-to-one function from sets X to Y mean that we have a onto function from Y to X. I can't understand it !! Someone can explain ??
I guess the "element of" symbol is ∈
belisarius
2010-10-14 19:05:21
`∈` and `` both work for me (and produce the same symbol), what browser are you using?
Zack
2010-10-14 21:40:01
I see rather than the symbol. I'm using Firefox 3.6.10.
GregS
2010-10-14 23:05:24
Must be a FF4 new feature then. Will change.
Zack
2010-10-15 01:13:12
@Zack `∈` is new in HTML5.
jleedev
2010-10-15 01:48:51
One-to-one does not mean bijective; it's actually a synonym for injective. So in this case, F may not have an inverse.
Jesse Beder
2010-10-15 02:33:25
@Jesse: I learned it as a synonym for bijective (and Wikipedia informs me that authors differ). But all injective functions have inverses, and those inverses are surjective; see edit.
Zack
2010-10-15 04:19:52
@jleedev: I fix it to be the actual Unicode characters. Hopefully that works better cross-browser.
Zack
2010-10-15 04:28:51
@Zack, every injective function has a *one-sided* inverse. An inverse, as usually defined, is such that fg = identity = gf, and a function has an inverse iff it is a bijection. (Also, I'm suspicious of wikipedia's statement that one-to-one function means bijective to some authors. I've never heard a mathematician use it that way (and wikipedia has no source). On the other hand, one-to-one correspondence is definitely a bijection.)
Jesse Beder
2010-10-15 04:53:10
Well, I don't claim to be a mathematician, but I have never before heard this strict definition of inverse - in fact, I thought even injectivity was too strong a condition. The definition I'm familiar with is, if `g(f(x)) == x` for all X *in the **image** of f* then g is the inverse of f.
Zack
2010-10-15 05:07:40
Er, "for all y = f(x) in the image of f", I meant.
Zack
2010-10-15 05:15:15
@Zack, read the wikipedia entry on inverses: http://en.wikipedia.org/wiki/Inverse_function. It reads pretty well (and notice that they use one-to-one as a synonym for injective).
Jesse Beder
2010-10-15 16:47:51
+2
A:
We can visualize this by drawing two circles, representing X and Y. The dots in the circle represent the elements in each set.
The arrows represent your function or "mapping".
So 1-1 means that every dot in the X circle maps to a unique dot in the Y circle.
Onto means that every dot has an arrow going to it. If you look at the picture, X is clearly not onto Y. There are two dots with no arrows coming in.
Now look at the "reverse" mapping by flipping the arrows on the lines.
Notice how in the reverse transform, every element of X has at least one element from Y going to it? That's the answer to your question. The 1-1 in the first picture (X to Y) means the second picture (Y to X) must be onto.
The wikipedia article on Surjective Functions explains this further.
nsanders
2010-10-15 01:26:50
Thanks for nice diagrams! in the second picture (the reverse) it's not a function from Y to X cause you have a free dot in Y set with no arrow going from it
Ams
2010-10-15 14:55:42
You are correct. The second diagram is not showing a function from the set of Y to the set of X.
nsanders
2010-10-16 23:00:02