I saw this definition and I got confused by it:
"Let E and F be two sets, which may or may not be distinct. A relation between a variable element x of E d a variable element y of F is called a functional relation in y if, for all x ∈ E, there exists a unique y ∈ 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 ∈ E the element y ∈ 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 thing that confused me in the definition above was the sentence "We give the name of function to the operation which in this way associates with every element x ∈ E the element y ∈ F which is in the given relation with x…" (he didn't define the word "operation")
In 1954, Bourbaki defined a function as a triple f = (F, A, B). Here F is a functional graph, meaning a set of pairs where no two pairs have the same first member, and he hasn't used the term "operation" which he hasn't defined in the first definition.
my problem with this definition is the fact that it does not resemble the notion of function as a process…
My questions are:
- why did he define in the first definition function as an operation (he didn't define what is an operation in the first place)?
- where the notion of function as a process appear in any of those two definitions?
Thank you for your patience and time!
The definitions appear in the following links, paper, and book:
https://en.wikipedia.org/wiki/History_of_the_function_concept
"Evolution of the Function Concept: A Brief Survey by
Israel Kleiner"
https://en.wikipedia.org/wiki/Function_(mathematics)
Nicolas Bourbaki – Set theory (book)
Best Answer
Maybe some more context will help...
See Elements of Mathematics: Theory of sets (Engl. transl.1968).
The "usual" mathematical object called relation in set theory is called by Bourbaki a graph, i.e. a set of ordered pairs [II.3.1].
A graph is said functional [II.3.4.: Def.9] when the "functionality" condition is satisfied.
What Bourbaki calls a relation is an expression of the language, i.e. an atomic formula based on a predicate symbols or a boolean combination, etc [see I.1.1, page 16 and Remark page 20: "intuitively terms represent objects and relations represent assertions"].
What is a functional relation (as defined in I.5.3)?
In a nutshell it is a formula $\varphi(x,y)$ satisfying the condition that:
Thus, a functional graph is a mathematical object while a functional relation is a linguistic object.
Wiki's quote above is the English translation of the text of the first French edition: Bourbaki (1939).
We can find into Archives de l'Association des Collaborateurs de Nicolas Bourbaki the corresponding "manuscript". See page 8:
If my conjecture is correct, the 2nd edition moved the "relation" name to the language of the theory and replaced it with "graph" and "correspondence" for the mathematical object.