BOOLEAN MATHEMATICS: GENERAL THEORY

Transcription

1 CHAPTER 3 BOOLEAN MATHEMATICS: GENERAL THEORY 3.1 ISOMORPHIC PROPERTIES The ame Boolea Arithmetic was chose because it was discovered that literal Boolea Algebra could have a isomorphic umerical aspect. Boolea Arithmetic is the correspodig umerical fudametal cocept to the well-kow Boolea Algebra. Whereupo logic may come to play a utmost role i the hardware, software ad firmware egieerig of moder Iformatio Sciece, similar to that played by Calculus i Physics. Nowadays, the devices of the techological hardware of itegrated circuits obey the priciples of Boolea Algebra. But i strict sese, these devices are cosequece of the great techological developmet of graphical logical gates ad/or, that is, the fudametal logical elemets with a isomorphic Boolea Geometry, which correspod to the basic operatioal cocepts of Boolea Algebra. For these reasos we ca say that Boolea Algebra, Boolea Arithmetic ad Boolea Geometry are the three mathematical aspects of the correspodet literal, umerical ad graphical laguages of a geeral Boolea Mathematics. Accordig to Whitesit [01], the mathematical coditios i which two systems are isomorphic are the followig: 51

2 1 st ) It is ecessary that for each operatio i oe system there is a correspodig operatio i the secod, although it may have a differet ame or symbol; d ) It is ecessary that the elemets of the two systems ca be paired i such a way that each elemet of oe system is paired with exactly oe elemet of the secod system, ad coversely. If both sets are fiite, this requiremet implies that each set has the same umber of elemets; 3 rd ) It is ecessary that this pairig ca be arraged i such a way that: If a "multiplicatio" table for ay operatio were costructed for oe system ad if each symbol "X" i the table were replaced by the symbol from the other system which is paired with this particular "X", the the resultig table would be the correct table for the correspodig operatio i the secod system. Accordig to Kohavi [0], two algebraic systems, each of them cosistig of a give set of elemets ad oe or more operatios, which satisfy a give set of postulates, are said to be isomorphic if the followig coditios are satisfied: For every operatio i oe system there exists a correspodig operatio i the secod system, although it may be deoted i a differet way. To each elemet " X i " i oe system correspods a uique elemet " Y i " i the secod, ad vice versa. Cosequetly, if both systems have fiite set of elemets, the they have the same umber of elemets. If i every postulate of the first system, each " X " is replaced by the correspodig " Y i ", ad every operatio is replaced by the correspodig operatio from the secod system, the the resultig postulate must be a valid oe for the secod system. I other words, two algebraic systems are isomorphic, if ad oly if, they are idetical except for the labels ad symbols used to represet the operatios ad elemets. Therefore, if two systems are isomorphic, the two systems are required to be structured i exactly the same way, except for the ames ad symbols used to describe the operatios ad elemets. The basic cocepts of Boolea Arithmetic, isomorphic to Boolea Algebra, are the followig: Boolea Arithmetical Variable (BAV) Boolea Variable; Boolea Arithmetical Fuctio (BAF) Boolea Fuctio; Boolea Arithmetical Equatio (BAE) Boolea Equatio; 5 i

3 Boolea Arithmetic came about because of the itroductio of the cocept of mathematical time i the area of mathematical thought, ad therefore Kat ( ) was restored as oe of the foreruers of Moder Mathematics [03]. The itroductio of the basic isomorphic cocepts from Boolea Algebra to Boolea Arithmetic i the search for abstractio is more suitable tha Gödel s Numbers, whe usig the umerical sequece of prime umber, as there is o eed to create ay arbitrary correspodece betwee literal ad umerical symbols. This search for abstractio i the coceptual pure mathematics was resposible for Russell s famous characterisatio of mathematics as the sciece i which we ever kow what we are talkig about, or whether what we say is true. The followig aalytical ad umerical cocepts were itroduced: Postulates formed by aalytical propositios, which iclude logical coectios betwee several variables i Techical Liguistics (also called, Boolea Liguistics) ad the isomorphic correspodet Boolea Equatio i Boolea Arithmetic; Oe set of postulates which correspods to the Simultaeous System of Boolea Equatios (SSBE); The correspodig Solvig Equatio of System of Boolea Equatios (SESBE) ad the correspodet isomorphic Solvig Equatio of System of Arithmetical Boolea Equatios (SESABE); THESIS (1 st PART of the deductio); THEOREMS ( d PART of the deductio); NUMERICAL SOLUTIONS (3 rd PART of the deductio); COMPLETE NUMERICAL TABLE of the solutios. Without ay arbitrary correspodece, it was possible to itroduce the same basic priciples of Boolea Algebra i the isomorphic Boolea Arithmetic. The, with the geeral solutio of ay Simultaeous System of Boolea Equatios (SSBE) that ca be see i Chapters 5 ad 6, we could obtai the complete umber of their deducig Theorems, Thesis, etc. I Chapter 11 this problem is preseted as a problem of Techical Liguistics where a Text Deductio is applied to a give set of Liguistic Postulates. The Boolea Arithmetical Fuctio (BAF) of a Boolea Fuctio has the followig coceptio: 1 st DEFINITION: BOOLEAN ARITHMETICAL FUNCTION (BAF). 53

4 Suppose it is give a Boolea Fuctio with "" variables, " Y = f (X,..., X, X 1 )". We call Boolea Arithmetical Fuctio (BAF) the correspodig isomorphic "Numerical Trasform" (NT) i a Boolea Arithmetical Field (BAFi), whose cardial umber is "" ad ordial umber is ω = ; X,..., XX1@. This "Numerical Trasform" (NT) is a terary of umbers give through a correspodig Truth Table Orderig formed by the sequece of the followig umerical parts: 1 st umerical part (also called "the abscissa of the NT"): It is a ordered strig of "^" bits i the biary represetatio, whose successio of these bits is obtaied from the Truth Table Orderig of the give fuctio. We ca see this abscissa o the right of the last colum of the TABLE 1), which should be read from dow to top; d umerical part: It is the cardial umber, represeted by the cardiality,"" of the Boolea Arithmetical Field (BAFi); 3 rd umerical part: It is the ordial umber, represeted by the ordiality, " ω = { X... XX1 }" of the Boolea Arithmetical Field (BAFi). The, this Numerical Trasform (NT) of the give Boolea Fuctio, has the followig otatio: " NT f1x, X,..., X, X 6 = α... α α α., X X... X X " < A S T H ^ H ^ 1... H H 1 NT {f} d α 0 = f(0,0,...0,0) d α 1 = f(0,0,...0,1) d α = f(0,0,...1,0) d α 3 = f(0,0,...,1,1) d ^ α = f(1,1,...,1,1) 1 TABLE. 1 "S" (from "SPACE"): Head of the colum (variables) "T" (from "TIME"): Name of the row (era of their values) The Boolea Arithmetical Variable (BAV) of a Boolea Variable has the followig coceptio: d DEFINITION: "BOOLEAN ARITHMETICAL VARIABLE (BAV) 54

5 As we ca see i TABLE 1, each colum i the cetral part which should be read from dow to top, represets a strig of " ^ " bits of a particular BOOLEAN ARITHMETICAL FUNCTION (BAF). This umerical strig is called the abscissa of a "BOOLEAN ARITHMETICAL VARIABLE (BAV), whose cardiality is "" ad ordiality is " ω ={ X... X X } Cosequeces: 1 ", i the same Boolea Arithmetical Field (BAFi). Give a set of "" Boolea Variables, C ={ X,..., Xi,..., X, X1 } i a Boolea Arithmetical Field (BAFi), whose cardiality is "" ad ordiality is "ω i ={ X... X... X X } 1 ", we have the followig Numerical Trasforms (NT) to the these ordered umerical variables: 1st. BAV: NT{ X } = [( 1 0 ) ].{ ; ω } = [( 10) ].{ ; ω } d. BAV: NT{ X } = [( 1 0 ) ].{; ω } = [( 1 0 ) ].{ ; ω }... ith. BAV: NT{ X i } = [( 1 0 ) ].{ ; ω }... th. BAV: NT{ X } = [( i 1 i 1 i ω = 1 0 ) ].{ ; } [ ].{ ; ω } It is opportue to observe that with these ew isomorphic cocepts of Boolea Arithmetical Variables (BAV) ad Boolea Arithmetical Fuctios (BAF), all the properties of Boolea Algebra are maitaied to the isomorphic Boolea Arithmetic. I this book, for the presetatio ad applicatio of these properties, we adopt a suitable symbolism [04] to the fudametal Boolea operatios: 1. Complemet: ( ). Boolea sum: ( & ) 3. Boolea product: ( v ) 55