Confused about set-theoretic definition of a function

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A function f: A -> B can be defined as an ordered triple of sets (A, B, X), where X consists of ordered pairs (a, f(a)) from A × B. The discussion clarifies that ordered tuples, such as pairs and triples, are not functions themselves but rather a way to represent relationships. The definition of ordered pairs as {{a}, {a, b}} is established, distinguishing them from functions. It is emphasized that the definition of functions using ordered pairs and triples is not circular. The clarification helps resolve the confusion regarding the set-theoretic definition of functions.
poochie_d
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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.
 
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poochie_d said:
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.
 
But aren't tuples other than the ordered pair defined as functions, so that the definition of functions as triples would still be circular?
 
poochie_d said:
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.
 
Oh I think I get it now. Thanks micromass!
 
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