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# 2 Set theoretical classes

### Sections

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 Set) may also be used as classes.

## 2.1 Creating set theoretical classes

• 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 Name) of the class created. The value bound to \in must be a function func which returns true if a GAP object belongs to C, and false 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 Objects and Elements in the GAP reference manual).

gap> FermatPrimes := Class(p -> IsPrime(p) and p = 2^LogInt(p, 2) + 1);
Class(in:=function( p ) ... end)


• View(class)

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.

gap> View(FermatPrimes);
Class(in:=function( p ) ... end)


• Print(class)

Print behaves very similarly to View, except that the defining information is being printed in a more explicit way if possible.

gap> Print(FermatPrimes);
Class(rec( in = function( p )
return IsPrime( p ) and p = 2 ^ LogInt( p, 2 ) + 1;
end))


• Display(class)

For classes, Display works exactly as Print.

• obj in class

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 obj.

• C1 = C2

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 IsIdenticalObj in the GAP reference manual).

• C1 < C2

The operation < for classes has no mathematical meaning; it only exists so that one can form sorted lists of classes.

## 2.2 Properties of classes

• IsEmpty(C) 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 C and should not be called directly by the user.

## 2.3 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.

gap> cmpl := Complement([1,2]);
Complement([ 1, 2 ])
gap> Complement(cmpl);
[ 1, 2 ]


• Intersection(list) F
• Intersection(C1, C2, ...) F

returns the intersection of the groups in list, resp. of the classes C1, C2, .... 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.

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) ])


• Union(C, D) F

returns the union of C and D.

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)
])


• 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.

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 ]


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CRISP manual
March 2016