2.9 Set Functions for Relations

Relations are not implemented as GAP domains, therefore the usual set functions (like Elements and Size) do not apply (even if we provide methods for them). However, union and difference of relations are implemented through the operators + and -.

rel1 + rel2

The operator + evaluates to the union of the relations rel1 and rel2 if both have the same degree.

    gap> a:= Relation( [ [  ], [ 4 ], [ 1, 4 ], [ 1 ] ] );;
    gap> b:= Relation( [ [ 1 ], [  ], [ 4 ], [  ] ] );;
    gap> a + b;
    Relation( [ [ 1 ], [ 4 ], [ 1, 4 ], [ 1 ] ] ) 

rel1 - rel2

The operator - evaluates to the difference of the relations rel1 and rel2 if both have the same degree.

    gap> a:= Relation( [ [  ], [ 4 ], [ 1, 4 ], [ 1 ] ] );;
    gap> b:= Relation( [ [ 1 ], [  ], [ 4 ], [  ] ] );;
    gap> a - b;
    Relation( [ [  ], [ 4 ], [ 1 ], [ 1 ] ] ) 

elm in rel

The operator in evaluates to true if the pair elm is in the relation rel, that is if elm[1] is related to elm[2], and to false otherwise. If elm is not a pair, or if an entry in pair exceeds the degree of rel an error is produced.

    gap> a:= Relation( [ [  ], [ 4 ], [ 1, 4 ], [ 1 ] ] );;
    gap> [1,2] in a; 
    false
    gap> [2,4] in a;
    true 

   
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