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Billy Rubina
Billy Rubina
00:34
@user21820 I often hear people saying that a "function" depends on the domain and codomain. These days I was doing some problems, isn't this "function that depends on domain and codomain" the graph of the function?
 
6 hours later…
user21820
user21820
06:35
@BillyRubina Unfortunately, the same people who make a lot of noise about functions and codomains often also are so unfamiliar with foundations of mathematics that they do not know what the concept of functions truly depends on. Here is one brief post that I wrote on the matter:
1
A: Why is it not sufficient to only check the third condition when verifying equality of functions?

user21820In foundations of mathematics, especially set-theoretic foundations, a function is nothing more than a certain kind of set of ordered pairs, from which you can easily extract its domain and its range, and there is no such thing as a codomain of a function. Two functions are equal exactly when the...

And note the comment I made there too:
If anyone wants a clear-cut reference, see "Set Theory" by Jech where he not only gives the same precise definitions in line with what I said here, and is careful to always say "is onto something" and never says "is a surjection", but also defines functions by giving only domain and mapping (not giving any codomain) (e.g. Theorem 3.2 and Lemma 3.3). — user21820 Nov 7 at 13:31
 
8 hours later…
Paramanand Singh
Paramanand Singh
14:59
@user21820: I read this concept of a function from Apostol's Mathematical Analysis. He defines an ordered pair as $(a, b) =\{a, \{a, b\} \} $ and says that a function $f$ is just any set of ordered pairs suhh that no two pairs have the same first member. First member of each pair of a function together form the domain. No need for codomain. But yes the notation $f:A\to B$ with codomain B is more or less standard.
 
5 hours later…
user21820
user21820
19:47
@ParamanandSingh Note that that notation does not in itself imply that the function has an intrinsic codomain. "f : S→T" is merely a statement relating f,S,T.

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