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% !TeX TS-program = pdftex % !TEX spellcheck =en_GB %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% classes.tex CRISP documentation Burkhard Höfling %% %% Copyright © 2000, 2011 Burkhard Höfling %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Chapter{Set theoretical classes} In {\CRISP}, a class (in the set-theoretical sense) is usually represented by an algorithm which decides membership in that class. Wherever this makes sense, sets (see "ref:Set") may also be used as classes. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Creating set theoretical classes}\null \index{classes!creation of} \>IsClass(<C>) C returns true if <C> is a class object. The category of class objects is a subcategory of the category `IsListOrCollection'. \>Class(<rec>) O \>Class(<func>) O returns a class <C>. In the first form, <rec> must be a record having a component `\\in' and an optional component `name'. The values of these components, if present, are bound to the attributes `MemberFunction' and `Name' (see "ref:Name") of the class created. The value bound to `\\in' must be a fu <func> which returns `true' if a {\GAP} object belongs to otherwise; cf. "MemberFunction" below. The second form is equivalent to `Class(rec(\\in := <func>))'. It is the user's responsibility to ensure that <func> returns the same result for different {\GAP} objects representing the same mathematical object (or element, in the {\GAP} sense; see "ref:Objects and Elements" in the {\GAP} reference manual). \beginexample gap> FermatPrimes := Class(p -> IsPrime(p) and p = 2^LogInt(p, 2) + 1); Class(in:=function( p ) ... end) \endexample \>View(<class>)!{for classes} If the class does not have a name, this produces a brief description of the information defining <class> which has been supplied by the user. If the class has a name, only its name will be printed. \begintt gap> View(FermatPrimes); Class(in:=function( p ) ... end) \endtt \>Print(<class>)!{for classes} `Print' behaves very similarly to `View', except that the defining information is being printed in a more explicit way if possible. \begintt gap> Print(FermatPrimes); Class(rec( in = function( p ) return IsPrime( p ) and p = 2 ^ LogInt( p, 2 ) + 1; end)) \endtt \>Display(<class>)!{for classes} For classes, `Display' works exactly as `Print'. \>`<obj> in <class>'{element test!for classes} \indextt{in!for classes}\relax \index{membership test!for classes}\relax returns true or false, depending upon whether <obj> belongs to <class> or not. If <obj> can store attributes, the outcome of the membership test is stored in an attribute `ComputedIsMembers' of \>`<C1> = <C2>'{equality!for classes} Since it is not possible to compare classes given by membership algorithms, two classes are equal in {\GAP} if and only if they are the same {\GAP} object (see "ref:IsIdenticalObj" in the {\GAP} reference manual). \>`<C1> \<\ <C2>'{comparison!for classes} The operation `\<' for classes has no mathematical meaning; it only exists so that one can form sorted lists of classes. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Properties of classes}\null \index{classes!properties of} \index{properties!of classes} \>IsEmpty(<C>)!{for classes} P This property may be set to `true' or `false' if the class <C> is empty resp. not empty. \>MemberFunction(<C>) A This attribute, if bound, stores a function with one argument, <obj>, which decides if <obj> belongs to <C> or not, and returns `true' and `false' accordingly. If present, this function is called by the default method for `\\in'. `MemberFunction' is part of the definition of directly by the user. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Lattice operations for classes}\null \index{lattice operations!for classes} \>Complement(<C>) O returns the unary complement of the class <C>, that is, the class consisting of all objects not in <C>. <C> may also be a set. \beginexample gap> cmpl := Complement([1,2]); Complement([ 1, 2 ]) gap> Complement(cmpl); [ 1, 2 ] \endexample \>Intersection(<list>)!{of classes} F \>Intersection(<C1>, <C2>, \dots)!{of classes} F \indextt{INTERSECTION\noexpand_LIMIT}\relax returns the intersection of the groups in <list>, resp. of the classes <C1>, <C2>, \dots. If one of the classes is a list with fewer than `INTERSECTION_LIMIT' elements, then the result will be a sublist of that list. By default, `INTERSECTION_LIMIT' is 1000. \beginexample gap> Intersection(Class(IsPrimeInt), [1..10]); [ 2, 3, 5, 7 ] gap> Intersection(Class(IsPrimeInt), Class(n -> n = 2^LogInt(n+1, 2) - 1)); Intersection([ Class(in:=function( N ) ... end), Class(in:=function( n ) ... end) ]) \endexample \>Union(<C>, <D>) F returns the union of <C> and <D>. \beginexample gap> Union(Class(n -> n mod 2 = 0), Class(n -> n mod 3 = 0)); Union([ Class(in:=function( n ) ... end), Class(in:=function( n ) ... end) ]) \endexample \>Difference(<C>, <D>) O returns the difference of <C> and <D>. If <C> is a list, then the result will be a sublist of <C>. \beginexample gap> Difference(Class(IsPrimePowerInt), Class(IsPrimeInt)); Intersection([ Class(in:=function( n ) ... end), Complement(Class(in:=function( N ) ... end)) ]) gap> Difference([1..10], Class(IsPrimeInt)); [ 1, 4, 6, 8, 9, 10 ] \endexample %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %E %% ¤ Dauer der Verarbeitung: 0.1 Sekunden (vorverarbeitet) ¤*© Formatika GbR, Deutschland |
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2026-03-28