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Old 2011-02-01, 19:57   #364
Mr. P-1
 
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Originally Posted by science_man_88 View Post
is this what you wanted ?
Not really, though I've had to spend quite a while thinking about what I do "want", why what you've written doesn't satisfy, and how to explain it to you. Plus I've been distracted...

Basically mathematicians try to express their ideas using a very limited vocabulary. The reason for this is so they can define very precisely how these terms interact, and how one can derive new statements (theorems) from previously proved theorems or unproven-but-assumed statements (axioms).

One consequence of this limited vocabulary as it applies to set theory is that it isn't necessary or even really possible to talk about "repeats". Given a set S and an object a, the only thing you can say about the latter's membership of the former is that either a\inS or a\not \inS. It isn't possible to even discuss how many times a\inS. The vocabulary doesn't permit it.

So what I "want" I guess, is for you to define these terms using the following vocabulary only:

Variables: a, b, c, X, Y, Z, etc.
Quantifiers: all, at least one, exactly one, no (as in no set)
Logical operators: and, or, not, neither ... nor, if ... then, iff (short for "if and only if".)
Set membership: member of
Punctuation: brackets and commas.

Other terms already defined using these terms.
Unambiguous synonyms, for example: "each", "every", instead of "all"; "implies" instead of "if ... then", etc.
Any additional words needed to make your sentences into grammatical English, so long as they don't carry substantive meaning.

For example. I can define "subset of" using the above by saying that "A is a subset of B" means the following:

For every x in A, x is in B.

Here "For" and "is" don't carry substantive meaning. (You could omit them and still understand the statement), "every" means "all"; "in" means "member of".

Another example: Define \empty:

For every x, x is not in \empty.

Binary union: A\cupB means:

For every x, x is in A\cupB iff (x is in A or x is in B)

Union of a collection: \cupC:

For every x, x is in \cupC iff x is in at least one X in C.

Do you think you could define binary intersection and intersection of a collection like this?
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Old 2011-02-02, 00:13   #365
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Quote:
Originally Posted by Mr. P-1 View Post
Not really, though I've had to spend quite a while thinking about what I do "want", why what you've written doesn't satisfy, and how to explain it to you. Plus I've been distracted...

Basically mathematicians try to express their ideas using a very limited vocabulary. The reason for this is so they can define very precisely how these terms interact, and how one can derive new statements (theorems) from previously proved theorems or unproven-but-assumed statements (axioms).

One consequence of this limited vocabulary as it applies to set theory is that it isn't necessary or even really possible to talk about "repeats". Given a set S and an object a, the only thing you can say about the latter's membership of the former is that either a\inS or a\not \inS. It isn't possible to even discuss how many times a\inS. The vocabulary doesn't permit it.

So what I "want" I guess, is for you to define these terms using the following vocabulary only:

Variables: a, b, c, X, Y, Z, etc.
Quantifiers: all, at least one, exactly one, no (as in no set)
Logical operators: and, or, not, neither ... nor, if ... then, iff (short for "if and only if".)
Set membership: member of
Punctuation: brackets and commas.

Other terms already defined using these terms.
Unambiguous synonyms, for example: "each", "every", instead of "all"; "implies" instead of "if ... then", etc.
Any additional words needed to make your sentences into grammatical English, so long as they don't carry substantive meaning.

For example. I can define "subset of" using the above by saying that "A is a subset of B" means the following:

For every x in A, x is in B.

Here "For" and "is" don't carry substantive meaning. (You could omit them and still understand the statement), "every" means "all"; "in" means "member of".

Another example: Define \empty:

For every x, x is not in \empty.

Binary union: A\cupB means:

For every x, x is in A\cupB iff (x is in A or x is in B)

Union of a collection: \cupC:

For every x, x is in \cupC iff x is in at least one X in C.

Do you think you could define binary intersection and intersection of a collection like this?
thank you for the lesson. I love the breakdown of language like floccinaucinihilipilification broken down it can be defined by the sum worthless + worthless +worthless + the act of. Using your examples as a guide I believe I'm up to the challenge. For every x, x is in A\capB iff (x is in A and x is in B). For every x, x is in \capC iff x is in all X in C . as you've probably seen in the theory on Mersenne primes thread I've been semi distracted as well. Needless to say , I've been trying to apply my understanding to my latest idea. This still hasn't completely helped me prove it and I've already been asked twice for the next Mersenne prime exponent.

Last fiddled with by science_man_88 on 2011-02-02 at 00:26
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Old 2011-02-02, 17:56   #366
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Quote:
Originally Posted by science_man_88 View Post
For every x, x is in A\capB iff (x is in A and x is in B).
Yes.

Quote:
Originally Posted by science_man_88 View Post
For every x, x is in \capC iff x is in all X in C .
Yes.

Quote:
Originally Posted by science_man_88 View Post
as you've probably seen in the theory on Mersenne primes thread I've been semi distracted as well. Needless to say , I've been trying to apply my understanding to my latest idea. This still hasn't completely helped me prove it and I've already been asked twice for the next Mersenne prime exponent.
We don't understand what you're saying. You make observations (which are, themselves, hard to follow) and then claim a link to Mersenne exponents without making the precise claim you're making explicit. If you wrote it as you did above (that is, unambiguously) it would go over much better.
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Old 2011-02-04, 12:01   #367
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Quote:
Originally Posted by CRGreathouse View Post
Yes.



Yes.



We don't understand what you're saying. You make observations (which are, themselves, hard to follow) and then claim a link to Mersenne exponents without making the precise claim you're making explicit. If you wrote it as you did above (that is, unambiguously) it would go over much better.
yeah but then I sound real book-like, my latest -> pairwise disjoint: a collection C is pairwise disjoint iff(for all A,B\in C A\cap B = \empty)

their's(pasted) ->
Quote:
A collection {Ci} of sets is
called pairwise disjoint if Ci ∩ Cj = βˆ… for all i, j with i =6 .
okay their's has one differnece to what is pasted it has the not equal sign and a j not the =6 .
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Old 2011-02-04, 12:15   #368
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Quote:
Originally Posted by science_man_88 View Post
yeah but then I sound real book-like.
That's to be commended!

If your statements are both correct and "book-like" then others will be able to understand you relatively easily. If they are wrong and book-like, it will be much easier to work out where you are going wrong and to provide corrections.

Paul
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Old 2011-02-04, 12:55   #369
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Quote:
Originally Posted by science_man_88 View Post
yeah but then I sound real book-like, my latest -> pairwise disjoint: a collection C is pairwise disjoint iff(for all A,B\in C A\cap B = \empty)

their's(pasted) ->
Do you see the difference between your definition and theirs?
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Old 2011-02-04, 12:58   #370
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Quote:
Originally Posted by CRGreathouse View Post
Yes.



Yes.



We don't understand what you're saying. You make observations (which are, themselves, hard to follow) and then claim a link to Mersenne exponents without making the precise claim you're making explicit. If you wrote it as you did above (that is, unambiguously) it would go over much better.

technically I think I could get that second one to align with: For all x, x is in \cap C iff(x is a one element subset of all X in C).
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Old 2011-02-04, 13:48   #371
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Do you see the difference between your definition and theirs?
they use i and j as indexes where mine looks to use full set names ?
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Old 2011-02-04, 13:51   #372
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Quote:
Originally Posted by science_man_88 View Post
they use i and j as indexes where mine looks to use full set names ?
No. The difference is that you have specified ANY sets, A and B which are elements of C.

They have excluded certain pairs of sets.

Which pairs have they excluded?
Why is that important?
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Old 2011-02-04, 13:54   #373
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Quote:
Originally Posted by Wacky View Post
No. The difference is that you have specified ANY sets, A and B which are elements of C.

They have excluded certain pairs of sets.

Which pairs have they excluded?
Why is that important?
oh doh I forgot that A!=B it's important because a set is always equal to itself and so no collection could be called pairwise disjoint under my definition, thanks for catching that.
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Old 2011-02-04, 14:18   #374
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oh doh I forgot that A!=B it's important because a set is always equal to itself and so no collection could be called pairwise disjoint under my definition, thanks for catching that.
Well... just to get nitpicky...

There are sets which are "sm-disjoint", that is, collections C for which all a,b in C are such that a ∩ b = {}. Can you find them?
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