Confused about set-theoretic definition of a function

1. Sep 15, 2011

poochie_d

I have read that a function f: A -> B can be defined as an ordered triple of sets (A,B,X), where X is the set of all ordered pairs $X = \{(a,f(a)) \in A \times B\}.$ But ordered tuples are really functions from $\{1, ..., n\}$ to (whatever set under consideration), right? So isn't this a circular definition? Or is there a more basic definition of functions that does not involve tuples? Any help would be greatly appreciated.

2. Sep 15, 2011

micromass

No, this is not true. The ordered tuple (a,b) is defined as {{a},{a,b}}. It's not defined as a function.

3. Sep 15, 2011

poochie_d

But aren't tuples other than the ordered pair defined as functions, so that the definition of functions as triples would still be circular?

4. Sep 15, 2011

micromass

No, triples can be defined as

$$(a,b,c)=((a,b),c)$$

And the definition of a function only uses ordered pairs and triples. So there is nothing circular.

5. Sep 15, 2011

poochie_d

Oh I think I get it now. Thanks micromass!