42.12 PreImages

PreImages( map, img )

In the first form PreImages returns the set of elements from the source of the mapping map that are mapped by map to the element img in the range of map, i.e., the set of elements elm such that img in Images( map, elm ) (see Images). map may be a multi valued mapping.

    gap> g := Group( (1,2,3,4), (2,4), (5,6,7) );;  g.name := "g";;
    gap> p4 := MappingByFunction( g, g, x -> x^4 );
    MappingByFunction( g, g, function ( x )
        return x ^ 4;
    end )
    gap> PreImages( p4, (5,6,7) );
    [ (5,6,7), (2,4)(5,6,7), (1,2)(3,4)(5,6,7), (1,2,3,4)(5,6,7), 
      (1,3)(5,6,7), (1,3)(2,4)(5,6,7), (1,4,3,2)(5,6,7), 
      (1,4)(2,3)(5,6,7) ]
    gap> p5 := MappingByFunction( g, g, x -> x^5 );
    MappingByFunction( g, g, function ( x )
        return x ^ 5;
    end )
    gap> PreImages( p5, (2,4)(5,6,7) );
    [ (2,4)(5,7,6) ] 

PreImages( map, imgs )

In the second form PreImages returns the set of all preimages of the elements in the set of elements imgs, i.e., the union of the preimages of the single elements of imgs. map may be a multi valued mapping.

    gap> PreImages( p4, [ (), (5,6,7) ] );
    [ (), (5,6,7), (2,4), (2,4)(5,6,7), (1,2)(3,4), (1,2)(3,4)(5,6,7), 
      (1,2,3,4), (1,2,3,4)(5,6,7), (1,3), (1,3)(5,6,7), (1,3)(2,4), 
      (1,3)(2,4)(5,6,7), (1,4,3,2), (1,4,3,2)(5,6,7), (1,4)(2,3), 
      (1,4)(2,3)(5,6,7) ]
    gap> PreImages( p5, [ (), (5,6,7) ] );
    [ (), (5,7,6) ] 

PreImages first checks in which form it is called.

In the first case it calls map.operations.PreImagesElm( map, img ) and returns this value.

The default function called this way is MappingOps.PreImagesElm, which runs through all elements of the source of map, if it is finite, and returns the set of those that have img in their images. Look in the index under PreImages to see for which mappings this function is overlaid.

In the second case if calls map.operations.PreImagesSet( map, imgs ) and returns this value.

The default function called this way is MappingOps.PreImagesSet, which returns the union of the preimages of all the elements of the set imgs. Look in the index under PreImages to see for which mappings this function is overlaid.

Previous Up Top Next
Index