Raul, about phi, remember the giant in Jack and the Beanstalk? When he said
Phee, Pheye, Phoh, Phum, he was trying to recite Greek letters! The symbol Ø
for the empy set looks vaguely like the Greek letter phi Ф , hence the name phi
for the empty set. You occasionally see the Greek capital phi used for the
empty set.
About the catechism, I apologize for not making clear its title should be
Catechism for Implementors
so Question 1 amounts to "how are you representing sets", just as you knew was
proper.
The questions where I supply answers give standard theoretical properties I
would like any implementation to have. I may have to settle for more than one
empty set, though, as J has empties of different shapes, and "shape boxing
elements" is what Match -: uses to determine whether two arrays "are the same".
I think I avoided this problem with my "set-marker" model, but I am
leaning
more now to the broader model of Dan and Fraser.
Kip
Raul Miller wrote:
> On Sun, Aug 2, 2009 at 7:23 PM, Kip Murray<[email protected]> wrote:
>> 1. Question: What is a set?
>> Answer:
>
> This question is too abstract, I think. A proper question
> would be "how are we representing sets", and this can
> depend on your application and on your universe of supported
> elements.
>
> For example, I have no problem with "a set is represented
> as a sorted list of unique boxes". But I also have no problem
> with "a set is represented as a sequence of bits marking
> which element in the universe is a member"
>
>> 2. Question: What is an element of a set?
>> Answer:
>
> See above.
>
>> 3. Question: When are two sets the same set?
>> Answer: Set H is set K provided each element of H is an element of K and
>> each element of K is an element of H.
>
> In the context of computer programs, this depends on how
> you represent sets. For both of my examples for question 1,
> -: would work. However, -: will not work for all possible
> representations of sets.
>
>> 4. Question: What is a subset of a set?
>> Answer: To say H is a subset of set K means H is a set, and each element
>> of H
>> is an element of K.
>
> Also, the details of how this is implemented can depend on how
> you represent sets. One fundamental approach
> would be to find the elements in the potential subset
> which do not appear in the potential superset -- if there
> are none the potential subset really is a subset.
>
> But if you represent sets with equal-length lists of bits
> you can use an expression which would not work for
> other potential approaches:
> 0 0 1 *./ .<: 0 1 1
> 1
>
>> 5. Question: What is an empty set?
>> Answer: The empty set, called phi, is the set which has no element.
>
> yes.
>
> But I never call the empty set "phi", and this name
> does not seem to be a common name for the empty
> set. Then again, I never call the empty set "george".
>
>> 7. Question: Why is the empty set a subset of every set?
>> Answer:
>
> This comes from the definition of subset (and of set):
>
> The set containing no elements is a set.
>
> For every set, if you removed all elements from the set,
> you would wind up with a set containing no elements.
>
>> 8. Question: Can the empty set be an element of a set?
>> Answer: That depends on 2, but the answer to 2 should make this answer
>> "Yes."
>
> Practically speaking, this depends on your application -- your
> "universe of consideration". If you have no use for empty sets
> in your sets, you should not waste any implementation effort
> on them.
>
> FYI,
>
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