Confused about set-theoretic definition of a function

[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.

 

[micromass]

But ordered tuples are really functions from $\{1, ..., n\}$ to (whatever set under consideration), right?

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

 

[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?

 

[micromass]

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

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.

 

[poochie_d]

Oh I think I get it now. Thanks micromass!