Timeline for How to define tuples?
Current License: CC BY-SA 2.5
17 events
| when toggle format | what | by | license | comment | |
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| Mar 18, 2011 at 16:53 | answer | added | Flash Sheridan | timeline score: 5 | |
| Jan 27, 2010 at 3:47 | comment | added | Joel David Hamkins | @Harry: Thanks, I didn't know that about Bourbaki. And I may have exaggerated about the 100 years; I'm not actually sure how long. But its been standard for many decades in set theory and logic, although less so recently in model theory (connections with algebra etc.). Surely the set-theoretic usage is often convenient: we might have a collection of functions and want to view them simultaneously with many different co-domains, without caring about invisible functors that pointlessly adjust the co-domains, just because somebody insists that every function must have a co-domain. | |
| Jan 26, 2010 at 21:44 | comment | added | Konrad Swanepoel | @Francesco: now I understand, thanks. | |
| Jan 26, 2010 at 20:29 | comment | added | Harry Gindi | @Joel: I know they don't carry much weight in the area of set theory, but in Bourbaki's first book on set theory, the one that was published in the 40s without any proofs, they do use the triple definition. I guess we can take this as more evidence that they were working in a proto-categorical setting. | |
| Jan 26, 2010 at 17:59 | comment | added | Andrea Ferretti | @Joel: sorry, I did not know this convention. I have always seen relations defined on fixed sets A, B, so relations have always been triples for me. | |
| Jan 26, 2010 at 16:49 | comment | added | Francesco Turco | @KonradSwanepoel: In the second definition you can avoid the dots by requiring a set X (intuitively made of x_1,...,x_n) prior to forming the n-tuple (x_1,...,x_n). So an n-tuple is just a function from the set of numbers from 1 to n (both included) to X. No dots at all. No need at all to name the elements of the n-tuple. | |
| Jan 26, 2010 at 15:51 | comment | added | Joel David Hamkins | @Andrea: In set theory, surjectivity is not a property of a function by itself, but a property of a function f and a set B: we say that a function f is onto B iff etc. (It doesn't make sense to say that a function, by itself, is surjective, unless B is understood or already specified; and usually it is.) And no, a relation is just a set of ordered pairs, so you can't get the codomain that way. Although I am of course aware that this terminology is different in other areas, but it is nevertheless completely standard in set theory (for about a hundred years). | |
| Jan 26, 2010 at 15:39 | answer | added | Leonid Dworzanski | timeline score: 1 | |
| Jan 26, 2010 at 15:02 | comment | added | Andrea Ferretti | You cannot tell the codomain? And then what does it mean that a function is surjective? Moreover a relation is a subset of the cartesian product of two sets, so it seems to me that you must include the codomain. A relation is a triple (A, B, I)... | |
| Jan 26, 2010 at 14:37 | comment | added | Joel David Hamkins | @Andrea: In set-theory, the word function is used universally to mean just a relation with the function property---you cannot tell the co-domain from it. | |
| Jan 26, 2010 at 14:30 | comment | added | Andrea Ferretti | By the way, your second definition seems to use functions, which require the use of triples (a function is a triple (A, B, f) where...) | |
| Jan 26, 2010 at 14:14 | answer | added | Harald Hanche-Olsen | timeline score: 34 | |
| Jan 26, 2010 at 13:12 | comment | added | Joel David Hamkins | What I mean is that, for example, every 3-tuple is explicitly also a 2-tuple. Every n-tuple is also a k-tuple for every k<n. | |
| Jan 26, 2010 at 13:09 | comment | added | Joel David Hamkins | Another issue with your first method is that the length of a finite tuple is not determined, when the x_i are themselves tuples, because you don't know how much to "unwrap" it. So this formulation is unsuitable, for example, in a context when one wants to consider the set of all finite tuples of a set of tuples. | |
| Jan 26, 2010 at 13:07 | answer | added | Tim Porter | timeline score: 2 | |
| Jan 26, 2010 at 11:09 | comment | added | Konrad Swanepoel | Why is your second definition more rigorous than the first? There are also dots.... Both definitions can be made rigorous using inductive definitions. In ZFC there are dozens of ways of constructing objects with certain properties. This is perhaps ugly, but it's very convenient. | |
| Jan 26, 2010 at 10:20 | history | asked | Francesco Turco | CC BY-SA 2.5 |