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*From*: Leslie Lamport <tlapl...@xxxxxxxxx>*Date*: Sun, 24 Aug 2014 07:49:57 -0700 (PDT)*References*: <4bfacee9-8875-40d3-99f0-08f55455b42c@googlegroups.com>

Hi FL,

This morning, the Wikipedia page you cite gives this quote from

Bourbaki in 1939:

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."

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.

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.

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.

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.

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

**Follow-Ups**:**Re: "Partial" Functions***From:*fl

**References**:**"Partial" Functions***From:*Leslie Lamport

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