Confusion about defn. of Surjective mapping in WIKI.

[ssd]

Reference: http://en.wikipedia.org/wiki/Bijection,_injection_and_surjection

Consider the two sets X & Y connected by a the relation y^2=x^2. (For simplicity we can take X={-2,2} and Y={-2,2}).Then can we call the mapping from X to Y to be surjective?
From the definition of WIKI, the answer appears to be 'yes'.

 

[micromass]

The relation that you mention does not give rise to a function. In your example, 2 is mapped to both 2 and -2. Thus 2 has multiple images. This is forbidden for a mapping.

 

[ssd]

The relation that you mention does not give rise to a function. In your example, 2 is mapped to both 2 and -2. Thus 2 has multiple images. This is forbidden for a mapping.

True that this is not a function. But my question is that WIKI's definition does not exclude it from surjective mapping.

 

[micromass]

Yes, it does exclude this as a surjective mapping. Wiki demands that a surjective mapping is a function. And since this is not a function, then this will also not be a surjective mapping...

 

[ssd]

Unfortunately, what I know is that "function" is defined through mapping. That is, definition of mapping comes before that of function, not the other way round.

 

[sponsoredwalk]

You know what a relation is right? If we have a relation S ⊆ ℝ'xℝ where
S = {(a,b)|(a ∈ ℝ') ⋀ (b ∈ ℝ)} then a function ƒ is the exact same, it's just a relation,
apart from one specific restriction we place on ƒ that distinguishes it from S.
A function has the property that if ƒ = {(a,b)|(a ∈ ℝ') ⋀ (b ∈ ℝ)} then a is the
first member of the tuple (a,b) in just one pair.

So, if A = {a,b,c} & B = {d,e,f} then S ⊆ A x B could be:

S = {(a,d),(a,e),(a,f),(b,d),(b,e),(b,f),(c,d),(c,e),(c,f)}

or

S = {(a,d),(a,e),(a,f),(b,f),(c,d),(c,e),(c,f)}

but ƒ ⊆ A x B is

ƒ = {(a,d),(b,d),(c,e)}

or

S = {(a,d),(b,f),(c,d)}

etc... I'm sure you see the distinction. I only wrote that stuff above because by thinking
along those lines I really don't see how you could get the impression that the wiki definition
allows for multiple elements of the co-domain to be mapped to multiple elements of the
domain (you know what I mean!):

A function is surjective (onto) if every element of the codomain is mapped to by at least one element of the domain.

My only guess is that when you read this sentence you missed the importance of the
inclusion of the word "by", but you think of it as saying that if 5 is an element of the
co-domain then if f(2) = 5 that's good but f(4) = 5 is also good for the definition of a
surjection, not so good for an injection.

Also:

A function can also be called a map or a mapping. Some authors, however, use the terms "function" and "map" to refer to different types of functions.
http://en.wikipedia.org/wiki/Function_(mathematics)

As I understand things at this present time the definition of a function is just that
of putting a restriction on a relation & does not arise out of the definition of a mapping,
they are mostly the same thing as far as I know, unless an author defines it differently.
I think that if you work with logic you assign an arity to your functions and relations &
this justifies mappings/functions of the form ƒ:ℝⁿ → ℝⁿ. I'll freely admit I thought a
mapping was of the form ƒ:ℝⁿ → ℝⁿ while a function was like f(x) = y (or ƒ:ℝ → ℝ)
but now I'm pretty sure that's just a ridiculous (or an unnecessary) distinction that comes
from being new to higher math & assuming some distinction.

Personally I prefer the definitions of injection/surjection/bijection given in
Sharipov's Linear Algebra & Multidimensional Geometry, check them out. My guess is that his
definition is more set oriented. I'd like to know why Sharipov defines it his way while 95%
of the rest of the texts do it the wiki way if anyone could shed some light on that.