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# Re: "Partial" Functions

Hi FL,

This morning, the Wikipedia page you cite gives this quote from
Bourbaki in 1939:

"Let E and F be two sets, which may or may not be distinct.  A
relation between a variable element x of E and a variable element y of
F is called a functional relation in y if, for all x \in  E, there
exists a unique y \in  F which is in the given relation with x.  "We
give the name of function to the operation which in this way
associates with every element x \in  E the element y \in  F which is in
the given relation with x, and the function is said to be determined
by the given functional relation.  Two equivalent functional relations
determine the same function."

The "for all x \in E" means that this is equivalent to what I regard
as the "ordinary" definition of a function with domain E, except that
for no good reason it introduces a set F that's a superset of the
range.  This is very strange, since it clearly fails to capture the
way mathematicians use the term function.  For example, mathematicians
generally consider the reals to be a subset of the complex numbers,
and therefore consider a real-valued function to be a special case of
a complex-valued function.  However, with Bourbaki's 1939 definition,
a real-valued function cannot be equal to a complex-valued function.
The passage you quoted doesn't give the actual definition in the 1954
edition, and it is consistent with that definition being the same as
the 1939 definition.  But, it is possible that they changed the
definition.  I don't have a copy of that volume of Bourbaki.

A formal definition of function that includes a superset of the domain
as one component, thereby introducing a formal notion of "partial
function", would not be inconsistent with informal mathematical
terminology.  For example, a mathematician say that "complex
functions" is the theory of complex-valued functions whose domain is
the set of complex numbers.  Those "functions" are actually partial
functions because they are undefined at their "singularities".  On the
other hand, they wouldn't consider a function on the real line to be
such a function, even though it is also a partial function with domain
the complex numbers.

In any case, I have never come across a rigorous definition of a
function in a book written by a mathematician (not a computer
scientist) other than simply a set of ordered pairs having different
first elements.  And I was educated as a mathematician.  Hence, my use
of the term "ordinary" definition.  If the current Bourbaki "Theorie
des Ensembles" introduces a concept of partial functions, then I will
abandon that term.  I hope someone who has a copy will tell us what it
says.

Leslie