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Spoonwood
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Name: Spoonwood
Birthday: 8/21/1979
Gender: Male
Interests: Fuzzy sets, mathematics, fuzzy numbers, fuzzy logic, neutrosophic sets, possibility theory, soft computing, type-2 fuzzy logic, hyperreal analysis, Doctor Who, Star Trek.
Expertise: Reversing the polarity of the neutron flow.
Occupation: Sonic screwdriver technician a
Industry: U.N.I.T.
Message: message me
Website: visit my website
Member Since:
12/14/2001
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| Consider the set of natural numbers N={0, 1, ...}. Suppose we have a member n of this set. There exists a subset {n_1, ..., n_n}of N which contains all factors of n. Let #{j} denote the number of members of j. A number c gets called composite iff #{c_1, ..., c_n}>2. A number p gets called prime iff #{p_1, ..., p_n}=#{1, p}=2. Two natural numbers m and p get called consecutive iff abs(m-p)=1 where abs stands for the absolute value function. A natural number o gets called odd iff there exists another natural number v such that 2v+1=o. Two odd numbers k and l get called consecutive odd numbers iff abs(o-l)=2.
Problems: Find a sequence of 5 consecutive natural numbers where each of them qualifies as composite. Find another sequence of 25 consecutive composite natural numbers. Show how one can find a sequence of n consecutive composite natural numbers for all n in N. Show that from a sequence of n consecutive composite natural numbers, one can always find a sequence of n+1 consecutive composite natural numbers. Find a sequence of 5 consecutive odd natural numbers where each of them qualifies as composite. Find another sequence of 25 consecutive composite odd natural numbers. Show how one can find a sequence of n consecutive composite odd natural numbers for all n in N. Show that from a sequence of n consecutive composite odd natural numbers, one can always find a sequence of n+1 consecutive composite odd natural numbers.
I'll post some answers in the comment section.
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| I believe (1) a very simple mathematical problem, but I doubt you can find it in many mathematics books. For some reason it fascinates me, even though I find it ever-so simple, and perhaps trivial. The questions in (2) perhaps indicates why I find such fascinating. Anyways, suppose the perspective of co-ordinate geometry. In other words, we can write shapes as equations and inequalities to figure out information about them.
(1) Given such a perspective, what does the product of two non-horizontal and non-vertical lines represent? If such a problem baffles you, then I'll ask you
(0) how does one write a line in terms of co-ordinate geometry?
(2) If you supply an answer for 1., what does the product of a triangle and another triangle represent? How did you multiply triangles? What conditions/do you have to put conditions do you have to put on multiplication of triangles in such a context? Does the result of multiplication of triangles change in terms of shape if you say rotate the triangle? If so, when and how does such a change occur? Does multiplication, or for that matter any other arithmetical operation make sense in this context? How do you make sense of such as an operation? Do we need to consider the ideas of addition, subtraction, multiplication, and division as relations? What does an arithmetic of shapes for circles and n-gons look like?
I'll supply (a few) answers of my own say in about a week, and if I don't, please remind me to do so.
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| By "Logic" in the title here, I just mean two-valued propositional logic. I just wanted a more concise title. I actually almost wrote this up about a week or so ago, but I lacked a detail. Then today I read in L. R. Lieber's Lattice Theory: The Atomic Age in Mathematics more or less the same idea. Really, one can draw more detail to such a picture if one likes.
A basic picture would have one circle below another circle with an F (for false) at the bottom circle and T (for true) at the top circle. One could add an arrow pointing from the F to the T for more detail. One can also draw a loop with an arrow from the F back to itself and from T back to itself for thoroughness. One could write this as F->T, and for more detail F->F, and T->T. Such arrows tell us that the truth value of false precedes (in the order-theoretic sense) the truth value of true, false precedes false, and truth precedes truth.
Edit: I've posted a picture in the photo section, the first two paragraphs just now describe the picture in words. Thanks very much to McScarry! I'd like to post it here in the post, but I don't think I can use that on this version of Xanga.
How does one derive the rest of propositional logic from this picture? We define the 'and' operation as the infimum (or greatest lower bound, denoted as inf) on a set of one or two element(s) from this picture, the 'or' operation as the supremum (or least upper bound, denoted as sup) on a set of one or two element(s) from this picture. Consequently,
F and F=F
F and T=T and F=F
T and T=T
F or F=F
F or T=T or F=T
T or T=T
Thus 'and' and 'or' here from this picture (or from F->T) behave the same way as they do in two-valued propositional logic, as those equations in tabular form give us the truth tables of two-valued logic.
How does one derive complementation? The complement on a lattice L (and we do have a lattice here) comes as an element c such that for an element 'a' we have BOTH sup{a, c}=sup L AND inf {a, c}=inf L satisfied. From the picture P, or from F->T we can see that sup P=T and inf P=F. Thus, for c to satisify the definition of a complement of T, we have to have both sup {T, c}=T and inf {T, c}=F. We only have inf {T, c}=F when c=F from the above. inf {T, F}=F also. Thus, T has complement of F, or c(T)=F for short. Similarly, for c to satisfy the definition of complement of F, we have to have both sup {F, c}=T and inf {F, c}=F. We only have sup {F, c}=T when c=T from the above also. sup {F, T}=T also. Thus, F has complement of T, c(F)=T.
And from there, the rest of two-valued propositional logic follows.
Perhpas more interesting comes as to consider multi-valued Boolean algebra logics say with truth values of a composite number ordered under divisibility (on the positive natural numbers). As an aside, logics which take their truth valued from prime numbers ordered under divisibility ALL come out as isomorophic to two-valued logic with F->T, as we have 1->P for any prime and that's the entire logical structure. As an example of such a mult-valued Boolean algebra logic, consider a logic L30 with truth values taken from the number 30 under divisibility. This logic has truth set {1, 2, 3, 5, 6, 10, 15, 30} with 1->2, 1->3, 1->5, 2->6, 2->10, 5->10, 3->15, 5->15, 6->30, 10->30, and 15->30 with the arrow meaning precedence in the order-theoretic sense above (I mean to imply that 5->30 and such also, but since we have a poset here, this comes as implied). In this case we can see that L30 has maximal truth value of 30, and minimal truth value of 1, with 2, 3, and 5 having a greater degree of truth value than 1. 2 has a lesser degree of truth than 10 also, while the truth values of 3 and 10 don't really work out as comparable, even though we still have sup {3, 10}=30 and inf {3, 10}=1. If we wanted to 'normalize' such a logic into the truth set [0, 1], I'd expect we'd want the number 1 in {1, 2, 3, 5, 6, 10, 15, 30} to have truth value of 0, and 30 to have truth value of 1. But how do we translate 2, 3, 5, 6, 10, and 15 into this interval and still have them behave similarly?
If one puts L30 as the degrees of membership that an element takes in a subset of a reference set, then we'd talk about a L-fuzzy set. One might contest this characterization from the standpoint that the degrees of membership need to take their values in an infinite membership set. However, one would probably call a set which has a membership function taking values in {0, .5, 1} a fuzzy set, even though we only have three values, because such a function doesn't take its values in {0, 1}.
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| Theorem: The system of truth values ({T, F}, implies) comes as equivalent to the system of truth values ({T, F}, and, or).
Scholia: The equivalence here comes as that used in lattice theory. So, specifically, I'll show that ({T, F}, implies) satisfies the poset definition of a lattice while ({T, F}, and, or) satisifies a*a=a (idempotence), a*b=b*a (commutavity), (a*b)*c=a*(b*c) (associativity), and a*(b@c)=a (absorption), where * and @ denote distinct operations. I'll denote implies by "->", and by '^', or by 'v'.
Proof: We have F->F, and T->T, thus for all members a, b, c of {T, F} a->a (reflexivity).
We only have a->b and b->a when a=b, since F->T and T->F do not both hold. Thus, a->b and b->a implies a=b (anti-symmetry).
If F->F, and F->F, then F->F. If F->F, and F->T, then F->T. If T->T, and T->T, then T->T. Thus a->b and b->c implies a->c.
T^T=T, F^F=F (idempotence) by duality idempotence holds for v.
F^T=T^F=F (commutavity) by duality commutavity holds for v.
F^(F^F)=F^F=F (F^F)^F=F^F=F
F^(F^T)=F^F=F (F^F)^T=F^T=F
F^(T^F)=F^F=F (F^T)^F=F^F=F
F^(T^T)=F^T=F (F^T)^F=F^F=F
T^(F^F)=T^F=F (T^F)^F=F^F=F
T^(F^T)=T^F=F (T^F)^F=F^F=F
T^(T^F)=T^F=F (T^T)^F=T^F=F
T^(T^T)=T^T=T (T^T)^T=T^T=T (associativity) by duality associativity holds for v.
F^(FvF)=F^F=F
F^(FvT)=F^T=F
T^(TvF)=T^T=T
T^(TvT)=T^T=T (absorption a^(avb)=a) by duality we have av(a^b)=a
Somewhat similarly one could state that ([0, 1], implies) where implies means 'less than or equal to' comes as equivalent to ([0, 1], max, min), thought one proves such a bit differently.
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| We would perhaps believe it a truism that there exists no most common use of numbers. After all, an engineer uses numbers differently than a mathematician than does a department score clerk. However, there does exist an exceedingly common use of numbers that almost everyone in most modern societies uses numbers for: to tell time.
Now, do we use these numbers to quantify time? In other words, if I talk about the time as 6:15 when my friend has a date at 7:00, do we use 6:00 to tell us how much time has passed since the meridian? We could and in rare cases people surely do so. However, generally, I suspect, we don't use them this way. Generally we use and think with numbers with respect to time, to structure how to order our lives. At 8:00 A. M. one eats breakfast, at 9:00 A. M. one starts work, at 7:00 P.M. the baseball game starts. This sort of use of numbers does NOT have to do with how much time has passed since the meridian, but rather how one time relates to another (one eats breakfast before work) and how certain times relate to certain behaviors of people (like how rush hour starts around 5:00 P. M.). On top of this, when one talks about quantities, one can say that quantity A has a smaller size than quantity B. But, do we say that 1:00 A. M. has a smaller size than 2:00 A. M.? No. I suppose we could say that 1:00 A. M. means that less time has passed since the meridian than if we talked about 2:00 A. M. However, do usually think so technically about time and say such things when we talk about time to other people? No. We usually we would just say 1:00 A. M. comes before 2:00 A. M. and leave things at that. And that implies that our most common (as much as we can get said to have a 'most common' use of numbers) use of numbers doesn't talk about numbers in terms of quantification, but rather in terms of order.
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