Images( map, elm )
In this form Images
returns the set of images of the element elm in
the source of the mapping map under map. 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> i4 := InverseMapping( p4 ); InverseMapping( MappingByFunction( g, g, function ( x ) return x ^ 4; end ) ) gap> IsMapping( i4 ); false # 'i4' is multi valued gap> Images( i4, () ); [ (), (2,4), (1,2)(3,4), (1,2,3,4), (1,3), (1,3)(2,4), (1,4,3,2), (1,4)(2,3) ] gap> p5 := MappingByFunction( g, g, x -> x^5 ); MappingByFunction( g, g, function ( x ) return x ^ 5; end ) gap> i5 := InverseMapping( p5 ); InverseMapping( MappingByFunction( g, g, function ( x ) return x ^ 5; end ) ) gap> Images( i5, () ); [ () ]
Images( map, elms )
In this form Images
returns the set of images of the set of elements
elms in the source of map under map. map may be a multi valued
mapping. In any case Images
returns a set of elements of the range of
map, either as a proper set (see Sets) or as a domain (see
Domains).
gap> Images( i4, [ (), (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> Images( i5, [ (), (5,6,7) ] ); [ (), (5,7,6) ]
Images
first checks in which form it is called.
In the first case it calls map.operations.ImagesElm( map, elm )
and returns this value.
The default function called this way is MappingOps.ImagesElm
, which
just raises an error, since their is no default way to compute the images
of an element under a mapping about which nothing is known. Look in the
index under Images to see how images are computed for the various
mappings.
In the second case it calls map.operations.ImagesSet( map, elms )
and returns this value.
The default function called this way is MappingOps.ImagesSet
, which
returns the union of the images of all the elements in the set elms.
Look in the index under Images to see for which mappings this function
is overlaid.
GAP 3.4.4