Man, haven't checked here in awhile

Five things:

(i) Post some new shit, this place has been dead for half a year!

(ii) You say, "+( d(RS) )=1" (I added extra spaces from parentheses)

I think it'd be better if you'd just stick with infix notation since formality is unnecessary, and that is especially so in ASCII. It's hard enough to read, don't make it harder! I don't get what you're trying to add here. Addition is a binary operation and you have three numbers. Let me refer to the less than, equal and greater than degrees by dl, de and dg, respectively. Since the two sets in the binary order are known, I'll omit them. You're asking to prove dl+(de,dg)=1. I don't get what the second pair is supposed to be. Did you not mean to throw in the extra parentheses and mean dl+de+dg? Since R and S are discrete values, I assume you mean singletons? It is straightforward because unlike intervals you cannot ever have overlap so only one of dl, de or dg will ever be true, and when it is true it is completely true (i.e., 1 degree). So you can just do it by cases when one is less than, equal or greater than the other number, and in each case we get 1+0+0=1, 0+1+0=1 and 0+0+1=1.

(iii) In the second to last paragraph you say this doesn't generalize to continuous real numbers. Do you mean to continuous intervals, since before you were dealing with discrete intervals?

I think you should first use the least upper bound (lub) and greatest lower bound (glb) when dealing with continuous intervals. It might even allow you to deal with cases that are not closed, unless you only want to work with compact sets of the real numbers. You also get a rather easy result from the real numbers (that I've probably had to prove four times this past year!) that if we have sets bounded sets A and B with B greater than all of A, then the sup(A) is less than inf(B). Though, when the sets overlap that does introduce some interesting properties for what is trying to be done here (producing fuzzy sets based on the real numbers).

(iv) Are you trying to generate a whole new object or just build something off of the real numbers?

I ask because your treatment toward the end reminds me of Dedekind cuts. If you haven't seen it, the basic notion is easy (proving the properties of the real number system are not so easy lol). You take Q and take all the ordered pairs (A,B) that partition Q such that A=(-∞,a) and B=[b,∞) or B=(b,∞). Basically rational numbers are the first B since the b that is the inf(B) is in Q. The second B has a b that is not in Q, but is still the inf(B) in this case. That would be the irrational number. In fact, that point that A and B "cut" at is the real number. Thus, every real number is an ordered pair (A,B) of one of these two types, and we get all the irrationals and rationals to "fill" in the gaps of Q to make R. Interestingly, you cannot prove properties about irrationals in this way (e.g., if something is transcendental). I'm loving my set theory class! But I digress.

It just makes me wonder what the goal of this is. You kind of just jump into this out of the blue and start attacking things. As one of my professors would say, you didn't set up the joke, and you're just throwing the punchline at us. He approaches all mathematics like that so everything has its punchline and the goal of a proof is to set it up. I feel like I missed the entire set up! Is it to build a "fuzzy metric" on R? Build "fuzzy numbers" entirely extended from R (like R is from Q with Dedekind cuts, e.g.). I feel lost.

(v) Have you looked into using and learning LaTeX? It's like -the- thing any math nerd has to know. There's also ports to Internet use which makes it easier to get symbolic material up to the web (Wikipedia, for instance, uses it). It's basically plain text with a processor that parses the fonts and graphics out of the text/commands to make the document the way you instructed. I'm going to do my math oriented blogs on my WordPress blog, after I get everything set up, eventually (http://www.bryangoodrich.com/blog ) because there's a free plug-in for it that allows me to easily input LaTeX stuff into my blog editor. Just something to look into for other blogs or find a way to use on this blog if you want to do symbolic stuff. It would certainly make the presentation of this material look infinitely better! At the very least, even if all you could do is post graphics in your blogs that are the image of the equations you produce with some sort of TeX, it would be better. Take a look. It's all free stuff.

(ii) should read the second to last sentence in paragraph five, i.e., "+(dl,(de,dg))=1" with my modification.

Note, also, another thing to consider, though very tedious (though, firefox might have some kind of add-on to make it easier to input them), there are a lot of symbolic things you can do with html tags. There's lists out there for what you can do to put in special characters (e.g., the copyright sign), and I even found out there's a sub/superscript tag, called "sub" and "sup" respectively, that makes all the text in those tags become, not surprisingly, superscript or subscript.

ASee if this works
BSubscript?

@bryangoodrich - 

[Addition is a binary operation and you have three numbers.]

This formula "+(d(RS))=1." in words reads "the sum of the degree that R is less than S, R is equal to S, and that R is greater than S equals 1." This doesn't so much consist of an addition, but an idea about a relationship here.

[Since the two sets in the binary order are known, I'll omit them. You're asking to prove dl+(de,dg)=1. I don't get what the second pair is supposed to be. Did you not mean to throw in the extra parentheses and mean dl+de+dg?]

I meant "+(d(RS))=1". It's not so much a sum to get done, but rather a relationship between all three in a way expressed as a summation.

[Since R and S are discrete values, I assume you mean singletons?]

No. If I did, I would mean real numbers, as {6}=6 here, which follows what I said above "If a1=a2, then we have I={x:a_1B gets computed by pairs of elements of A and B... i. e. {1, 2} 1/2 {1, 3}, since 3>2 and the other three possibilities don't hold for '>'. In other words {1, 2} is 1/2 less than {1, 3}, 1/4 equal to {1, 3} and 1/4 greater tan {1, 3}.

[It is straightforward because unlike intervals you cannot ever have overlap so only one of dl, de or dg will ever be true, and when it is true it is completely true (i.e., 1 degree). So you can just do it by cases when one is less than, equal or greater than the other number, and in each case we get 1+0+0=1, 0+1+0=1 and 0+0+1=1.]

As I recall that case seemed pretty simple and straightforward to me... but that's not what I meant. I meant any finite discrete set of real numbers.

[In the second to last paragraph you say this doesn't generalize to continuous real numbers. Do you mean to continuous intervals, since before you were dealing with discrete intervals?]

Notice I let points qualify as intervals. My first couple of examples dealt with discrete sets of point such as {1, 2, 3} and {2, 6, 7}. If we have two discrete sets of intervals, such as {1, [3, 5], 6} and {pi-3, [1, 4], 9} we'll end up with the same sort of ordering problem since we'll end up with infinites there also. In other words... I didn't cover all discrete intervals... I covered a set of discrete numbers like {1, 3, 5}... but if you start talking about a set of continuous numbers [1, 4], then my first procedure doesn't work.

[Though, when the sets overlap that does introduce some interesting properties for what is trying to be done here (producing fuzzy sets based on the real numbers).]

As I recall, I wasn't producing fuzzy sets based on the real numbers. I didn't have any fuzzy numbers there, just sets of real numbers. I did have fuzzy relations though, as the degree of membership that A<B for {1, 2, 3}=A and {3, 6, 7}=B ends up as a fuzzy relation.

[Are you trying to generate a whole new object or just build something off of the real numbers?]

I think both.

[I ask because your treatment toward the end reminds me of Dedekind cuts. If you haven't seen it, the basic notion is easy (proving the properties of the real number system are not so easy lol). You take Q and take all the ordered pairs (A,B) that partition Q such that A=(-∞,a) and B=[b,∞) or B=(b,∞). Basically rational numbers are the first B since the b that is the inf(B) is in Q. The second B has a b that is not in Q, but is still the inf(B) in this case. That would be the irrational number. In fact, that point that A and B "cut" at is the real number. Thus, every real number is an ordered pair (A,B) of one of these two types, and we get all the irrationals and rationals to "fill" in the gaps of Q to make R. Interestingly, you cannot prove properties about irrationals in this way (e.g., if something is transcendental). I'm loving my set theory class! But I digress.]

I certainly don't see how you see a similarity. I don't mean to cut the real numbers into rational and irrational numbers.

[It just makes me wonder what the goal of this is. You kind of just jump into this out of the blue and start attacking things. As one of my professors would say, you didn't set up the joke, and you're just throwing the punchline at us. He approaches all mathematics like that so everything has its punchline and the goal of a proof is to set it up. I feel like I missed the entire set up!]

It's to order sets of real numbers, as the title reads "ordering sets of real numbers." We order real numbers simply by ''. With sets of real numbers, such as [1, 3] and [2, 4] I basically intend to help to provide some sort of method to order [1, 3] as partailly less than [2, 4], partially equal to [2, 4], and partially greater than [2, 4]. Hopefully in a way that seems consistent with the numbers as magnitudes of some sort.

[I feel like I missed the entire set up! Is it to build a "fuzzy metric" on R? Build "fuzzy numbers" entirely extended from R (like R is from Q with Dedekind cuts, e.g.). I feel lost.]

I think your difficulty lies in that you've looked at this from R instead of P(R). There exists quite a difference here. I dealt with sets of real numbers. You can't get them from R, since [1, 3] does not belong to R. [1, 3], however, does belong to P(R).

I didn't discuss fuzzy numbers... but you can't build them from R, as I understand them. You have to deal wtih P(R) and functions from R into [0, 1], since fuzzy numbers come as sets of real numbers with a degree of membership given by their membership functions. Real numbers can get said to have an indicator function and all qualify as singletons... but since you can let {r}=r for any real number, and the indicator function always equals 1, you can ignore indicator fucntions, and also ignore P(R) and just use R instead.

Basically we can order singeltons just like real numbers, since we have {r}=r. We don't need anything we don't know already here. But, when we start ordering sets of real numbers like {1, 3, 5} and [2, 6], we have a different story. It seems natural that we end up with fuzzy relations, since 3 belongs to {1, 3, 5} and 4 belongs to [2, 6] and 32. So, it seems that {1, 3, 5} is partially less than and partially greater than [2, 6]. The question becomes how much for each relation and how do we do this systematically (if I remember my original intention correctly, which I think I do)? Well, I didn't deal with such an example like that one now, did I? I think it reasonable to say {1, 3, 5}=A is mostly less than [2, 6]=B (since if we assume a uniform distribution of the numbers and take a member a from A and a member b from B, most of the time a<b), a little bit greater than [2, 6], and only equal to [2, 6] at three points.

@Spoonwood - 

FYI, the reason I changed the notation was because comments will parse out the brackets. You might want to check over your comment and fix where content was cut out (e.g., your first statements).

d(rs)=1 instead of +(d(RS))=1... except I want the sum sign at the beginning of the expression.

Let "dl" stand for the "degrees less than" relationship and "dg" stand for the "degrees greater than" relationship. You said,

"+(dl(R,S), (d(R=S), dg(R,S))=1"

I'm not following why the latter two statements d(R=S) and dg(R,S) are grouped together. Extra set of parentheses? You're basically saying +(X,Y)=1 where X=dl(R,S) and Y=(d(R=S), dg(R,S)). I was assuming you're saying that the "sum" of these needs to equal 1, otherwise we'd have a trichotomy violation of the real numbers, essentially. So your original statements about r and s were just for arbitrary real numbers, and your exercise was for interval numbers? You'd prove it much the same way because it follows from the trichotomy which applies to all real numbers, and R and S are just collections of real numbers. I might be able to write out a formal proof of that if you want, in pdf format. Essentially, if you have it for any arbitrarily chosen r and s in their respective domains, and it's already proven in general for such arbitrary r and s, then you just show by cases what has to hold about r and s from R and S respectively. Since it's arbitrarily chosen, it applies universally. Therefore, showing it for the r and s case for each possible R and S case is your proof. That's the outline.

Have you looked into Hasse diagrams? It might be useful, and help characterize the kind of relationships you're depicting (because they add a convenient way to summarize a lot of relationships). It is much more difficult to envision and deal with every ordered pair in a relationship, since when you say A={1,2} and B={1,3}, we're really comparing (1,1), (1,3), (2,1) and (2,3) from AxB to how much they belong to each of the lower, equal and greater than relations on AxB. Hasse diagrams present the lattice structure for ordered relations (I believe you'd have to do one for each relation, since they'd be ordered structures of the form (AxB,L/E/G) depending on the relation).

As for discrete real numbers, I think you might mean finite sets? Discrete values are generally whole numbers or something translatable to whole numbers (because technically whatever one wants to do with integers you can translate it in some way to reduced forms of rationals I believe). I see what you're getting at though. The two major differences with these intervals are that some can be continuous (e.g., all real numbers between [0,1]) or finite sets. I just thought by discrete you might be referring to singletons {n} for n being some real number and trying to show about "dl+de+dg=1" which would be trivial to show as I pointed out in my misinterpretation earlier.

I don't get your terminology by discrete interval, because something like {0,1,I,5} where I=[50,51] does not work out as "discrete" by any standard. It's continuous, or at least has a continuous "element" in it. I mean, think about this set being a domain to a function. You'd have it defined over I, but with isolation points 0, 1 and 5. You don't have a discrete interval since this "domain" is only discrete on its isolation points, which excludes I. But you are correct, the method of proof for the finite interval cannot carry over to continuous intervals. It would still be proven through the trichotomy property, but I'm not sure how to go about it since end points (lubs and sups) can be problematic, and overlaps can be problematic. My intuition would be that an argument through lubs and sups would be able to "separate" overlap appropriately, but then that would only work for single relations. To show it for the other two relations and relate those results such that dl+de+dg=1 is going to be more difficult. I cannot see the proof at this time. I'll think about that one though. I don't think it'd be too much more difficult than the finite case (usually generalization proofs like that from finite to infinite are either almost trivial or insanely more difficult. I believe and hope it's the former lol)

Fuzziness is really whenever there isn't a whole value for some property of a set. The relations to become fuzzy, and they are sets, so they are generating a kind of fuzzy set, which is what I was getting at. What I was wondering is if the goal (it's good to know where we're going with this manipulation to the real numbers) is to adjoin some kind of fuzziness to the real numbers, either in a sort of "metric" sense or by generating an entirely new "fuzzy" structure extended from the reals (which is not necessarily an exclusive goal).

I certainly don't see how you see a similarity.

That's not what I was getting at. The similarity is that you're imposing a new sort of set which introduces new properties held between these new objects generated by dicing up the real numbers and relating them in such and such a way (by their fuzzy ordering relations to each other). It doesn't, however, cut things up in necessarily unique or important ways. Dedekind cuts produced something--a point at which the cut exists which gives us rational and irrational numbers generically. From those cuts defined the way they are, we can construct the real numbers. Without a clear goal in mind I don't see what is going forward with this programme, but the whole dl(A,B), etc. is taking reals and imposing a new kind of structure or property, one where "dl+de+dg=1" holds. Whether it is fruitful, it is an interesting investigation onto the reals.

You say your goal is to "order the real numbers" but that's a bit redundant. They're already ordered! lol What it looks like is taking the trichotomy property of the real number systems (based on one ordering) and generalizing it to all intervals in a manner which produces fuzzy relations between intervals (or numbers/singltons) based upon the structure of the real number system. I haven't dealt with infinite orderings in my set theory class yet, and it is going to be rather complicated from what it looks like. I might not be able to fruitfully help with this for another month at least!

I do have to think about this as P(R) instead of R, but the problem with that jump, though not necessarily a bad thing, is that in some regard you lose the properties of R that are very, very good. Technically they still exist, because in analysis you're still looking at intervals (subsets) of R, so you're still dealing with the same "critter" as you might have as a member of P(R), but that is the only significant difference: one is a subset of R and the other is a member of P(R). Whether I look at A=[0,1] and B=(1,4] as a subset of R or a member of P(R), I still know from analysis that B contains upper bounds of A, B is bounded, so infA, infB, supA, supB exist, and that, for LE being "less than or equal to", (infA)LE(supA)LE(infB)LE(supB).

[I'm not following why the latter two statements d(R=S) and dg(R,S) are grouped together.]

I think I couldn't group all three together. I tried that in the comment and it didn't work, so I think I did that in the original instead.

[So your original statements about r and s were just for arbitrary real numbers, and your exercise was for interval numbers?]

Pretty much.

[You'd prove it much the same way because it follows from the trichotomy which applies to all real numbers, and R and S are just collections of real numbers.]

I don't follow. For {1, 3, 5}=A and {1, 4, 6}=B we don't have either AB.

[I don't get your terminology by discrete interval, because something like {0,1,I,5} where I=[50,51] does not work out as "discrete" by any standard. It's continuous, or at least has a continuous "element" in it.]

It's definitely not continuous in the real numbers, since .5 belongs to the reals and it doesn't belong to the set. So, I call it discrete. This seems neither here nor there though.

[The similarity is that you're imposing a new sort of set which introduces new properties held between these new objects generated by dicing up the real numbers and relating them in such and such a way (by their fuzzy ordering relations to each other).]

No, I don't have a new sort of set. It's a new sort of relation between sets, as neither {1, 3, 6} and {1, 2, 5} consists of new sorts of sets... but having a relation of which doesn't follow trichotomy comes as new.

[Whether it is fruitful, it is an interesting investigation onto the reals.

You say your goal is to "order the real numbers" but that's a bit redundant.]

I said "It's to order sets of real numbers, as the title reads "ordering sets of real numbers." Or "Ordering SETS of real numbers" if you like.

[What it looks like is taking the trichotomy property of the real number systems (based on one ordering) and generalizing it to all intervals in a manner which produces fuzzy relations between intervals (or numbers/singltons) based upon the structure of the real number system.]

I don't see how I've generalized trichotomy. If I had, then A>B, A=B, or Ab, a<b still holds if we have singletons.

[Technically they still exist, because in analysis you're still looking at intervals (subsets) of R, so you're still dealing with the same "critter" as you might have as a member of P(R), but that is the only significant difference: one is a subset of R and the other is a member of P(R).]

Perhaps it's only a single difference, but it still has a signficance.

@bryangoodrich - 

See above.

@Spoonwood - 

Like I've said, you need to use different notation in your comments because using less than and greater than get parsed as html tags, so some statements in your comment are disappearing I believe (e.g., sentence 3).

What you seem to have is a new structure (P(R), dl,de,dg) based on the fact that "dl+de+dg=1." It allows us to take subsets of R and relate them according to the property that the sum of degrees satisfying the trichotomy property of R for elements in any X in P(R) is equal to 1. It is an interesting structure since it does introduce an element of fuzziness to consider, and extends from the real numbers. The next step would be to see what else we can add to the structure and see what the consequences are (e.g., adding other metrics or operations such as addition or multiplication or both). The generalization of the trichotomy is that you've generalized it "fuzzily" for subsets of the reals.

As for nomenclature, the issues is that you don't have a fully discrete set if you're including intervals in the set. An interval needs to be either continuous or discrete I believe, but you have sets which can contain continuous or discrete intervals. I mean, we could have the infinite set of whole numbers as a subset of R, or sets like {0,2,I,9} where I is some continuous interval. To call that set discrete just doesn't seem accurate, nor required. It obfuscates things. It may be neither here nor there, but when you classify something as, say, discrete, you're predicating certain properties on it. We want to make sure those properties are clearly defined and accurate. That is all.

@bryangoodrich - 

[The next step would be to see what else we can add to the structure and see what the consequences are (e.g., adding other metrics or operations such as addition or multiplication or both).]

Well, I guess we could start with the operations for interval arithmetic.
[a1, a2]+[b1, b2]=[a1+b1, a2+b2]
[a1, a2]-[b1, b2]=[a1-b2, a2-b1]
[a1, a2]*[b1, b2]=[min(a1*b1, a1*b2, a2*b1, a2*b2), max(a1*b1, a1*b2, a2*b1, a2*b2)]
[a1, a2]/[b1, b2]=[a1, a2]*[1/b2, 1/b1], 0 does not belong to [b1, b2]

If A={a1, a2, ... an} and B={b1, b2, ..., bn} have the same number of elements and we have them ordered a1<a2<...<a_n, b1<b2<...<b_n, then we can probably just add them pairwise... e.g. {3, 4, 6}+{5, 7, 8}={3+5, 4+7, 6+8}={8, 11, 14}. Actually, no, I don't like that. Since A could mean any a_n in A, and B also mean any b_n, we'd really need to add all pairs together... so {3, 4, 6}+{5, 7, 8}={3+5, 3+7, 3+8, 4+5, ...}={8, 9, 10, 11, 12, 13, 14}. We'll probably want to do the same thing for the other arithmetic operations.

I think it more obfuscates things to call {1, 3, [4, 6], 7} continuous than to call it discrete.

@Spoonwood - 

You could just call it neither since "continuous" and "discrete" don't really predicate on that set. Just call it a set.

Thinking about it, we really cannot create a structure off the dl, de, and dg relations because if we stick to standard constructions we'd want to just have the ordered pairs, call them l, e and g respectively. Then we're simply tacking on a measure d which takes the degree, relative to the set in question. I'll have to think about it further.

@bryangoodrich - 

[Thinking about it, we really cannot create a structure off the dl, de, and dg relations because if we stick to standard constructions we'd want to just have the ordered pairs, call them l, e and g respectively.]

We don't have to stick to standard two-valued constructions. In ohter words, can have a "structure", but a fuzzy structure here. We can have fuzzy relations where we have degrees of membership, such as ({4, 6}, .5). Or ({4, 6), .2). The difference here comes as that given the pair ({a_1, ..., a_n}, {b_1, ..., b_n}) we should be able to calculate the fuzzy relation for dl, de, and dg. E. G. for ({1, 3}, {2, 4}) we can write calculate dl as 3/4 (since ({1, 2) <) holds, ({1, 4}, <) holds and so on), and dg as 1/4. We can then write
({1, 3}, {2, 4}, dl=3/4, de=0, dg=1/4). We can extend the notion here to the idea of a structure in general, as long as we permit the existence of fuzzy relations for the notion of structure. We can't deny the existence of fuzzy relations in applied mathematics, so why deny them in pure mathematics?

@Spoonwood - 

It wasn't an issue about membership, though it would be interesting to see how one constructs a fuzzy structure and what it could lead to, but the issue was the very definition of the signature. The dl, de and dg are not sets, they were values with a property. We would be using standard relations of l, e and g, and then have a statement in the language such that dl+de+dg=1 that needs to be satisfied, i.e., all models of this structure would require that they satisfy this statement.

@bryangoodrich - 

[It wasn't an issue about membership, though it would be interesting to see how one constructs a fuzzy structure and what it could lead to, but the issue was the very definition of the signature. The dl, de and dg are not sets, they were values with a property. We would be using standard relations of l, e and g, and then have a statement in the language such that dl+de+dg=1 that needs to be satisfied, i.e., all models of this structure would require that they satisfy this statement.]

No, dl, de, and dg are not sets. However, they do qualify as fuzzy relations. One could have the structure (P(R), dl, de, dg) where P(R) indicates a crisp set, and dl, de, and dg fuzzy relations.

Problem: Given that A denotes [a_lower, a_upper] and B denotes [b_lower, b_upper] and a_lower comes as less than b_lower and a_lower does not equal a_upper and b_lower does not equal b_upper, find the degrees to which A comes as less than B, A equals B, and A comes as greater than B.

Find the intersection of A and B, A^B=J. Find the union of A and B, AUB=O Find the subset S of A where each s of S comes as less than j of J. Find the subset K of A such each s of S belongs to J. Take the width of S, J, K, and O, w(S), w(J), w(K), w(O). w(S)/w(O)=degree to which A comes as less than B, w(J)/w(O) indicates the degree to which A=B, w(K)/w(O) indicates the degree to which A comes as greater than B.

E.G. for [1, 4], [2, 6], we have J=[2, 4], O=[1, 6], S=[1, 2], K=[2, 4], w(J)=2, w(O)=5, w(S)=1, w(K)=2. So, A comes as less than B to 1/5 degree, equal to B to 2/5 degree, and greater than B to 2/5 degree.

K=O in general, so the degrees that A comes as greater than B and the degree that A=B come out equal.

[But, how do we order sets with partial overlap, such as C={1, 2, 3} and D={2, 3, 4} by '<'?

Well, we know 12, so C is 1/9 greater than D. Here the pair (C, D) has degree of 2/3 in <. ]

This sort of method would mean that for A={1, 3, 5} and B={1, 3, 5} we have that A comes as less than* B to 1/3 degree, A=*B to 1/3 degree, and A greater than* B to 1/3 degree. Except, A=B. So, such relations as less than*, =*, and greater than* characterizes the members of the sets, and not the sets.