Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
International Journal of Enhanced Research Publications, ISSN: XXXX-XXXX Vol. 2 Issue 4, April-2013, pp: (1-4), Available online at: www.erpublications.com On Infinitesimal Sets and Hyper Real Numbers Tahir H. Ismail Department of Mathematics, College of Computer Sciences and Mathematics, University of Mosul, Mosul 41002, Iraq Abstract: In this paper, we introduce and define a new type of numbers called hyper real numbers and infinitesimal sets as a new type of sets in the set of real numbers β .We use these concepts to define and characterize a relation between infinitesimal sets and hype real numbers. Keywords: Non Standard Analysis, Infinitesimal, Hyper Real Numbers Introduction Through this paper we need the following definitions and notations: Definition 1.1 [6],[4],[2] Areal number Ο is called unlimited if its absolute value is larger than any standard integer.so a nonstandard integer Ο is also unlimited real number, Ο+1/2 is an example of unlimited real number that is not integer. Definition 1.2 [5],[8],[9] A real number x is called infinitesimal if its absolute value is smaller than infinitesimal. 1 π for any standard number n, of course 0 is Definition 1.3 [2] A real number x is called limited if x is not unlimited. Definition 1.4[3],[7] Areal number x is called appreciable if x is neither unlimited nor infinitesimal. Definition 1.5 [1],[3] Two real numbers x and y are infinitely near, denoted by x β y if x-y is infinitesimal. The axioms of IST is the axioms of ZFC together with three additional axioms which are called the transfer axiom, the idealization axiom and the standardization axiom .They are as follow: Transfer Axiom [6]: for each standard formula F(x,t1,t2,β¦,tn) with only free variables x,,t1,t2,β¦., t n , the following statement is an axiom βπ π‘ π‘1 , π‘2 , β¦ β¦ t n (βπ π‘ πΉ(π₯, π‘1 , π‘2 , β¦ . , π‘π ) βΉ βπ₯ πΉ( π₯, π‘1 , π‘2 , β¦ β¦ . , π‘π )) Idealization axiom [6]: For each standard formula B(x,y), with free variables x,y the following is an axiom βπ π‘πππ Ξβπ₯ βπ¦ πΞ β§ π΅(π₯, π¦) βΊ βπ₯ βπ π‘ π΅(π₯, π¦) Standardization Axiom [6] For every formula A(z) internal or external, with free variable z, the following is an axiom π π‘ π π‘ βst x π’ βπ¦ βπ§ (zΟ΅ Y βΊ zΟ΅X β§ A(Z)) Definition 1.6 [3],[9] If x is limited real number, then it is infinitely near to unique standard real number called the standard part of x, denoted by 0x . Definition 1.7 An infinitesimal set, denoted by β is an external convex additive subgroup of β. Page | 1 International Journal of Enhanced Research Publications, ISSN: XXXX-XXXX Vol. 2 Issue 4, April-2013, pp: (1-4), Available online at: www.erpublications.com Definition 1.8 A hyper real number is the algebraic sum of a real number and an infinitesimal set. The following remark gives us some properties and example of infinitesimal set and hyper real numbers. Remark 1.9 1. If β1 is an infinitesimal set, a is a real number and Ξ± is a hyper real number, then Ξ± β‘ a+β1 β‘{a+ x:x ββ1 }, β1 is the infinitesimal part of the hyper real number Ξ±. 2. The unique internal infinitesimal sets in β are {0}, but there are many external infinitesimal sets, the monad of zero. 3. Every infinitesimal set is a hyper real number. On the other hand, there are hyper real numbers which are neither real number nor infinitesimal set as in the following example. Example 1.10 If Ο is unlimited positive real number, then the following sets are all hyper real numbers, which are neither real numbers nor infinitesimal sets 1. 1+β β‘ { 1+π βΆ ππ β } 2. Ο +L β‘ { Ο + t : t β L } Definition 1.11[3],[5] An infinitesimal set β2 is sub infinitesimal set of the infinitesimal set β1, if β1 ββ2 and denoted by Max(β1,β2 )= β1. Remark 1.12 1) The sum of two infinitesimal sets β1 and β2 is defined by β1+ β2=max(β1, β2). 2) The product of two infinitesimal sets β1and β2 is defined by β1β2 β‘ {x y:x ββ1 and yββ 2}. 3) The product of an infinitesimal set β1 with any appreciable real number leaves β1 invariant. 1 4) If an infinitesimal set does not contain 1, then by external induction it does not contain ( ) for any standard integer n π and is thus included in β. Definition 1.13 If a real number P does belong to an infinitesimal set β1 ,then β1 π is an infinitesimal set included in β called the relative infinitesimal set of a hyper real number Ξ± β‘ P + β 1. The hyper real number Ξ± β‘ a + β 1 can be written in the following β β form Ξ±β‘ ( 1 + π1 ) ,and the hyper real number ( 1 + 1 ) is called module of a .which is exactly the collection of real π numbers that leave Ξ± invariant undermultiplication. Proposition 1.14 If Ξ± β‘ a +β 1 , Ξ² β‘ b + β2 are two hyper real numbers b ββ2 and a ββ 1 . Then 1) Ξ± + Ξ² β‘ a +b + β1β 2 β‘ a+ b +max (β1, β2 ) 2) πΌ β π½ β‘ π β π + πππ₯ (β1 ,β2 ) 3) Ξ± . Ξ² β‘ a b +max (π β 2 , b β 1, β 1β 2 ) 4) 1 π½ 1 β‘ πΌ π π½ π 5) β‘ ( 1+ π + β2 π 1 ) (max (π β2 , b β 1, β 1β 2 ) π2 Proof: (1), (2) and (3) are obvious and direct Proof. (4): We have 1 π½ β‘ 1 π+β2 β β‘ 1 β π (1+ 2 ) π 1 β‘ ( 1 π 1+β2 ) (1) π Since 2 is the relative infinitesimal set contained in β 2 , we have π number, therefore β2 π β β , this means that β2 π is infinitesimal real Page | 2 International Journal of Enhanced Research Publications, ISSN: XXXX-XXXX Vol. 2 Issue 4, April-2013, pp: (1-4), Available online at: www.erpublications.com 1 (1+ β2 ) π β2 β )(1+ 2 ) π π β (1+ 2 ) π (1+ β‘ β2 β‘ (1 + π ) From (1) we get 1 1 β2 π π β‘ (1 + π½ ) Proof (5): From (4) we have β 1 β2 π π β‘ (π +β1 )( (1 + π½ 1 πβ2 π π β‘ (π + β‘ πΌ π +β2 + β1 β2 π 1 πβ2 +πβ1 +β1 β2 + ( π π 1 π π2 β‘ + )) π ) ) (π β2 + π β1 +β1 β2 ) Therefore, β π½ πΌ 1 π π2 β‘ + max(π β2 , π β1 , β1 β2 ) Remark 1.15 From the above proposition we have the following 1. If β is a hyper realnumber, then (β ββ) is not equal to zero but to β1 , that is the infinitesimal part of β which contain zero. 2. If ββ‘ π +β1 is hyper real number, then is not equal to 1, but is either equal to module of β or to 1 + 2 β1 , β β where c is real number. 3. If β β‘ πΌ +β1 and π½ β‘ π +β2 are two hyperreal numbers and suppose that β1 contains β2 , and that there exists a real x belonging to both β and π½ . β π Since β is invariant, it can be translated by an element of β1 , πππ π ππππ π½ isinvariant, it can be translated by an element of β2 , it follows thatββ‘ π₯ +β1 πππ π½ = π₯ +β2 . This means that β contains π½. So we have the following proposition. Proposition 1.16 If β and π½ are two hyperreal numbers .Then either β and π½ are disjoint or one contains the other. Proof: Suppose that ββ‘a+β1 and π½ β‘ π +β2 are two hyper realnumbers, we have to prove that either β and π½ are disjoint or one contains the other. If β and π½ are disjoint, then no one of them contains the elements of the other and we have nothing to prove. Now if β and π½are not disjoint, this means that ββ‘x+β1 and π½ β‘ π₯ + β2 Then either π©1 ββ2 or β2 β π©1 . Hence ββ π½ or π½ ββ. References [1]. I.v. Berg, βExtended use of IST,β Annals of pure and Applied logic, vol. 58, pp.73-92, 1998. [2]. M. Davis, Applied Nonstandard Analysis, New York, John Wiley & Sons, 1977 . [3]. F. Diener and M. Diener, Nonstandard Analysis in Practice, Springβverlag , Berlin, Heidelberg, 1995 . [4]. F. Diener and G. Reeb, Nonstandard Analysis, Herman, 1989. [5]. I.O. Hamad, βA new type of Infinitesimal in Nonstandard Analysis,β Zanko Journal of pure and Applied Science, Salaheddin University ,Vol.16 ,No.1, 2004. [6]. E. Nelson, βInternal Set theory: A new approach to Nonstandard Analysis,β Bull. Amer. Math. Soc.,Vol.83, No.6, pp. 11651198, 1977. [7]. A. Robinson , Nonstandard Analysis, 2ed, North-Holland, Pup. Com., Amsterdam, 1970. [8]. K.D. Stroyan and W.A. Luxembourg, Introduction to the Theory of Infinitesimal, New York , Academic Press, 1976. [9]. N.Vokil, Real Analysis through modern infinitesimal, Cambridge University Press, 2011. Page | 3