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April 20th, 2010, 10:52 AM   #1
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Set Theory : Closures

Hey guys,
I got a general question. Does any attribute of a set (like symmetric,reflexive and etc) have a closure?
If they don't, then how do I prove that there's no asymmetric closure, for example?
Thanks in advance.
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April 20th, 2010, 11:51 AM   #2
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Re: Set Theory : Closures

I'm going to give a particular interpretation to your question; you'll have to tell me if it's right.

You're considering relations over sets, and attributes of those relations. A relation over a set is a subset of pairs with members in the set; an attribute is a subset of the class of relations. You're looking for a relation which is a superset of a given relation and has the attribute. (One could also consider unique closures, as in your examples: no superset of the original relation can have the attribute without containing all members of the new relation.)

A natural counterexample is the attribute of unicorn-ness. No relations are unicorns, so there is no "unicorn closure". Perhaps this is trivial, and we should consider nonempty attributes (attributes for which relations exist which have that attribute).

So consider the attribute of falsity, which includes only the empty relation. (A relation FALSE with this property is such that a FALSE b is false for every a, b in the set.) Any nonempty relation has no 'falsity' closure.


Clearly, for an attribute A to have a closure it must contain the relation TRUE which contains every pair of members in the set (else that relation has no A-closure). Any such attribute has an A-closure -- if nothing else, map every relation to TRUE.

So perhaps the question should be, which attributes have unique closures? Certainly some attributes exist which have closures but no unique closures. Consider an underlying set has an odd number of elements (at least three) and the attribute oddness which is true only for relations with an odd number of members. The relation FALSE contains 0 members, so it does not have oddness. But adding any pair to the relation gives it oddness, and so there is no unique oddness closure. But there are an odd number of elements and hence an odd number of pairs, so the relation TRUE that contains all pairs has oddness.


Thanks for the question, that was fun. I hadn't thought about any of this before.
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April 21st, 2010, 03:46 AM   #3
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Re: Set Theory : Closures

Yes, you've understood my question precisely but your examples weren't clear enough to me. Like the first one, what is uni-corn attribute? I google it but didn't find any declaration or an addition explanation? Also, I am not quiet sure what is this FALSE relation you've mentioned. For example, I got A={1,2,3,4}. so how does FALSE work here?

I really appreciate your time and efforts, it sounds very interesting. THANK u very much
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April 21st, 2010, 05:25 AM   #4
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Re: Set Theory : Closures

Quote:
Originally Posted by fed2black
Yes, you've understood my question precisely but your examples weren't clear enough to me. Like the first one, what is uni-corn attribute?
True if the relation is a unicorn and false otherwise.
http://images.pictureshunt.com/pics/u/unicorn-1846.jpg

A relation is a set, a collection of pairs. A unicorn is an animal with one horn. No relations are unicorns.

Quote:
Originally Posted by fed2black
Also, I am not quiet sure what is this FALSE relation you've mentioned. For example, I got A={1,2,3,4}. so how does FALSE work here?
Let me first give < on this set. < is {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)}, because for < is true for (1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4) and false for all other pairs on A. For example, we write 1 < 3 because (1, 3) is in <, while 4 < 2 is false because (4, 2) is not in <.

FALSE is {} because FALSE is true for no pairs of A. For example, 4 FALSE 2 is false because (4, 2) is not in {}.
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April 21st, 2010, 10:36 AM   #5
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Re: Set Theory : Closures

Quote:
Originally Posted by CRGreathouse
True if the relation is a unicorn and false otherwise.
http://images.pictureshunt.com/pics/u/unicorn-1846.jpg

A relation is a set, a collection of pairs. A unicorn is an animal with one horn. No relations are unicorns.
I feel the need to point out that this is a great example of a property of a relation.
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April 21st, 2010, 11:11 AM   #6
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Re: Set Theory : Closures

LOL!!! I am an ass! Anyway, thanks, your example is good, but I think I have something more simple.
For example, I have noticed there are symmetric, transitive and reflexive closure. I am sure that there's a good
example of R that doesn't have a asymmetric closure or anti-transitive closure. but somehow I can't figure one.
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April 21st, 2010, 12:23 PM   #7
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Re: Set Theory : Closures

Quote:
Originally Posted by fed2black
I am sure that there's a good
example of R that doesn't have a asymmetric closure or anti-transitive closure. but somehow I can't figure one.
TRUE over any nonempty set has no asymmetric closure: it contains (a, a), so all supersets of it contain (a, a), and no relation containing (a, a) can be asymmetric. This goes through with antitransitivity since antitransitive relations are also asymmetric.
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