What is the formal Dirichlet-Bourbaki definition of a function?

I have come across this in this essay: http://www.k-12prep.math.ttu.edu/journal/contentknowledge/meel01/article.pdf on page 1.

I know what a function is and I can write down a definition. What I would like to know is what the the definition is that is specifically known as the formal Dirichlet-Bourbaki definition.


A function $f:X\rightarrow Y$ is a subset of $X\times Y$ such that $(x,y_1)\in f$ and $(x,y_2)\in f$ implies $y_1=y_2$.

See for example here: http://books.google.com/books?id=8Wn3SJDIhWwC&pg=PA121&lpg=PA121&dq=dirichlet-bourbaki+function&source=bl&ots=TO_fvKoRy2&sig=RfPmjmSYVJpbeRBOMMadPD8EUNg&hl=en&sa=X&ei=KOxUUbjSIuiaiQLIwIGYBA&ved=0CG4Q6AEwCA#v=onepage&q=dirichlet-bourbaki%20function&f=false

  • $\begingroup$ Thanks for the answer. So you are saying that this definition is what is know as the formal Dirichlet-Bourbaki definition? $\endgroup$ – Thomas Mar 29 '13 at 1:12
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    $\begingroup$ Yeah, see here: books.google.com/… $\endgroup$ – Elchanan Solomon Mar 29 '13 at 1:20
  • $\begingroup$ there are many ways to state the same thing. it appears that dirichlet never gave a formal definition of function, while bourbaki gave more than one $\endgroup$ – suissidle Mar 29 '13 at 1:22
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    $\begingroup$ You left out $\forall x\in X\;\exists y\;(\;(x,y)\in f).$ $\endgroup$ – DanielWainfleet Apr 7 '16 at 17:55

The term "Dirichlet-Bourbaki definition of a function" appears to be a term used by some primary/secondary-level mathematics educators for the contemporary set-theoretic notion of a function. Here is some information about the definitions of Dirichlet and Dedekind, excerpted from Israel Kleiner's article Evolution of the Function Concept: A Brief Survey. See the full article (free) for much more on the complex history.

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  • $\begingroup$ Thanks for this. This is very helpful! $\endgroup$ – Thomas Mar 29 '13 at 4:21

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