TITLE: Does a bijection between to sets always permit a well defined function
QUESTION [0 upvotes]: Almost all definitions of bijections that I have come across define bijections as functions. I am not sure if I am wording this correctly, but does a bijection between two sets (where I mean that each element in both sets is matched with exactly one element in the other) always allow for a well defined function? In other words, if we know there is a bijection between two sets in the manner defined above, can we be assured that there has to be a function between the two sets?

REPLY [1 votes]: bijections $\subset$ functions $\subset$ relations.
So, yes, every bijection, by definition is a function.
Relation:  a group of pairings from set $A$ to members of set $B$.  Not all elements need to be paired. And elements from either set may be in multiple pairs (or not in any pairs).
function: a relation in which every element of $A$ is paired with exactly one element of $B$.  However not every element of $B$ needs to be paired and the elements of $B$ may be in multiple pairs.  But every element of $A$ is in one and only one pair.  But two different elements of $A$ may map to the same element of $B$.
surjection: a function in which every element of $B$ is in a pair.  This is called "onto" because the function maps "onto" every element of $B$.
injection: a function in which the each element of $A$ is mapped to a different element of $B$.  This is sometimes called 1-1 because every element of $A$ is mapped 1-1 to distinct elements of $B$.  However this is ambiguous as it doesn't mean every element of $B$ has an element mapped to it.  In my opinion, I do not like to call this 1-1, and I do not.
bijection:  a function that is both a surjection and an injection.  i.e. every element of $A$ is map to exactly one distinct element of $B$ and all elements of $B$ are mapped to.  This is sometimes called "1-1 and onto" or sometimes just 1-1.  In my opinion, this should be the correct use of 1-1; I do not like to use 1-1 to describe functions that are only injective but not surjective.
BUT
what does "well-defined" mean?
It's possible to have a function/mapping that simply can not be described completely.  Say there is a 1-1 and onto mapping from the real numbers onto the real numbers so that every real number is mapped to a real number but the mapping is completely arbitrary and has no pattern or predictability.  The only way to describe it is to simply list all the mappings but that's impossible as the reals are uncountable.
Is that a function?  Absolutely.  No question.
Is it "well-defined"? ..... well, that depends on what the definition of "well-defined" is.  "well-defined" usually means "unambiguous" or "mathematically correct".  "Let $f(x)$ be the largest real number less than $x$" is not well-defined because there is no largest real number less than $x$".  "Let $f(x)$ = $a + b$ where $x = a*b$" is not well defined because there are many different $a,b$ so there are many different $a + b$; the function is ambiguous.
But the arbitrary mapping is neither mathematically impossible, nor ambiguous.  It is merely.... indescribable.  Does that count as not "well-defined"?  In my opinion, I'd say it is well-defined but others my, rightfully, disagree with me.
=== post script ====
In response to a comment:  I do wish to point out that the "indescribable" function which I ...er... described.... is in itself, not "defined" at all.  Probably 70% to 90% of all mathematician will disagree with me on this but I do not consider being "defined" as a requirement for being "well-defined".  I view "well-defined" to be more as a statement of "conceptually sound".  $f(a*b) = a+b$ is not conceptually sound (as $f(12) = 3+4 = 2+6$) and so is not well-defined.  Any bijective mapping between to sets is sound and is a function, whether we can define it or not.  Basically I do not consider "defining" a function to be relevant to the nature of the function and I consider the function's existence to be a form of definition-- we can look at it.
But, no, it is very possible to have a function that simply can not be defined in words.
tl;dr
Any bijection between sets is always a function.  It may not be able to be described but it is a function.
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