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NAME

       sofs - Functions for Manipulating Sets of Sets

DESCRIPTION

       The  sofs  module  implements operations on finite sets and relations represented as sets.
       Intuitively, a set is a collection of elements; every element belongs to the set, and  the
       set contains every element.

       Given  a set A and a sentence S(x), where x is a free variable, a new set B whose elements
       are exactly those elements of A for which S(x) holds can be formed, this is denoted B = {x
       in  A  :  S(x)}. Sentences are expressed using the logical operators "for some" (or "there
       exists"), "for all", "and", "or", "not". If the existence of  a  set  containing  all  the
       specified elements is known (as will always be the case in this module), we write B = {x :
       S(x)}.

       The unordered set containing the elements a, b and c is denoted {a, b, c}.  This  notation
       is  not  to  be confused with tuples. The ordered pair of a and b, with first coordinate a
       and second coordinate b, is denoted (a, b). An ordered pair  is  an  ordered  set  of  two
       elements.  In  this  module  ordered  sets  can  contain  one,  two  or more elements, and
       parentheses are used to  enclose  the  elements.  Unordered  sets  and  ordered  sets  are
       orthogonal, again in this module; there is no unordered set equal to any ordered set.

       The set that contains no elements is called the empty set. If two sets A and B contain the
       same elements, then A is equal to B, denoted A = B. Two ordered sets  are  equal  if  they
       contain the same number of elements and have equal elements at each coordinate. If a set A
       contains all elements that B contains, then B is a subset of A. The union of  two  sets  A
       and  B  is  the  smallest  set  that contains all elements of A and all elements of B. The
       intersection of two sets A and B is the set that contains all elements of A that belong to
       B.  Two  sets  are  disjoint if their intersection is the empty set. The difference of two
       sets A and B is the set that contains all elements of A that  do  not  belong  to  B.  The
       symmetric  difference  of  two  sets is the set that contains those element that belong to
       either of the two sets, but not both. The union of a collection of sets  is  the  smallest
       set  that contains all the elements that belong to at least one set of the collection. The
       intersection of a non-empty collection of sets is the set that contains all elements  that
       belong to every set of the collection.

       The  Cartesian  product of two sets X and Y, denoted X x Y, is the set {a : a = (x, y) for
       some x in X and for some y in Y}. A relation is a subset of X x Y. Let R  be  a  relation.
       The  fact  that  (x,  y)  belongs  to R is written as x R y. Since relations are sets, the
       definitions of the last paragraph (subset, union, and so on) apply to relations  as  well.
       The domain of R is the set {x : x R y for some y in Y}. The range of R is the set {y : x R
       y for some x in X}. The converse of R is the set {a : a = (y, x) for some (x, y) in R}. If
       A  is  a  subset of X, then the image of A under R is the set {y : x R y for some x in A},
       and if B is a subset of Y, then the inverse image of B is the set {x : x R y for some y in
       B}.  If  R  is  a  relation from X to Y and S is a relation from Y to Z, then the relative
       product of R and S is the relation T from X to Z defined so that x T  z  if  and  only  if
       there exists an element y in Y such that x R y and y S z. The restriction of R to A is the
       set S defined so that x S y if and only if there exists an element x in A such that x R y.
       If S is a restriction of R to A, then R is an extension of S to X. If X = Y then we call R
       a relation in X. The field of a relation R in X is the union of the domain of  R  and  the
       range  of R. If R is a relation in X, and if S is defined so that x S y if x R y and not x
       = y, then S is the strict relation corresponding to R, and vice versa, if S is a  relation
       in  X,  and  if  R is defined so that x R y if x S y or x = y, then R is the weak relation
       corresponding to S. A relation R in X is reflexive if x R x for every element x of  X;  it
       is  symmetric  if  x R y implies that y R x; and it is transitive if x R y and y R z imply
       that x R z.

       A function F is a relation, a subset of X x Y, such that the domain of F is equal to X and
       such  that  for every x in X there is a unique element y in Y with (x, y) in F. The latter
       condition can be formulated as follows: if x F y and x F z then y = z. In this module,  it
       will  not be required that the domain of F be equal to X for a relation to be considered a
       function. Instead of writing (x, y) in F or x F y, we write F(x) = y when F is a function,
       and  say  that  F  maps  x  onto  y, or that the value of F at x is y. Since functions are
       relations, the definitions of the last paragraph (domain,  range,  and  so  on)  apply  to
       functions as well. If the converse of a function F is a function F', then F' is called the
       inverse of F. The relative product of two functions F1 and F2 is called the  composite  of
       F1 and F2 if the range of F1 is a subset of the domain of F2.

       Sometimes,  when  the  range of a function is more important than the function itself, the
       function is called a family. The domain of a family is called the index set, and the range
       is  called  the  indexed set. If x is a family from I to X, then x[i] denotes the value of
       the function at index i. The notation "a family in X" is used for such a family. When  the
       indexed  set is a set of subsets of a set X, then we call x a family of subsets of X. If x
       is a family of subsets of X, then the union of the range of x is called the union  of  the
       family x. If x is non-empty (the index set is non-empty), the intersection of the family x
       is the intersection of the range of x. In this module, the  only  families  that  will  be
       considered  are families of subsets of some set X; in the following the word "family" will
       be used for such families of subsets.

       A partition of a set X is a collection S of non-empty subsets of X whose union  is  X  and
       whose elements are pairwise disjoint. A relation in a set is an equivalence relation if it
       is reflexive, symmetric and transitive. If R is an equivalence relation in X, and x is  an
       element  of  X,  the  equivalence  class  of  x  with respect to R is the set of all those
       elements y of X for which x R y holds. The equivalence classes constitute  a  partitioning
       of  X.  Conversely,  if  C  is  a partition of X, then the relation that holds for any two
       elements of X if they belong to the same equivalence class,  is  an  equivalence  relation
       induced  by  the partition C. If R is an equivalence relation in X, then the canonical map
       is the function that maps every element of X onto its equivalence class.

       Relations as defined above (as sets of ordered pairs) will from now on be referred  to  as
       binary relations. We call a set of ordered sets (x[1], ..., x[n]) an (n-ary) relation, and
       say that the relation is a subset of the Cartesian product X[1] x ... x X[n] where x[i] is
       an  element  of X[i], 1 <= i <= n. The projection of an n-ary relation R onto coordinate i
       is the set {x[i] : (x[1], ..., x[i], ..., x[n]) in R for some x[j] in X[j], 1 <=  j  <=  n
       and  not  i  =  j}.  The  projections  of  a  binary  relation R onto the first and second
       coordinates are the domain and the range of R respectively. The relative product of binary
       relations  can  be  generalized  to  n-ary  relations as follows. Let TR be an ordered set
       (R[1], ..., R[n]) of binary relations from X to Y[i] and S a binary relation from (Y[1]  x
       ...  x  Y[n])  to Z. The relative product of TR and S is the binary relation T from X to Z
       defined so that x T z if and only if there exists an element y[i] in Y[i] for each 1 <=  i
       <=  n  such  that  x  R[i]  y[i] and (y[1], ..., y[n]) S z. Now let TR be a an ordered set
       (R[1], ..., R[n]) of binary relations from X[i] to Y[i] and S a subset of  X[1]  x  ...  x
       X[n].  The multiple relative product of TR and S is defined to be the set {z : z = ((x[1],
       ..., x[n]), (y[1],...,y[n])) for some (x[1], ..., x[n]) in S and for some (x[i], y[i])  in
       R[i],  1  <=  i  <= n}. The natural join of an n-ary relation R and an m-ary relation S on
       coordinate i and j is defined to be the set {z : z = (x[1], ..., x[n], y[1], ...,  y[j-1],
       y[j+1],  ...,  y[m])  for  some (x[1], ..., x[n]) in R and for some (y[1], ..., y[m]) in S
       such that x[i] = y[j]}.

       The sets recognized by this module will be represented by elements of the  relation  Sets,
       defined as the smallest set such that:

         * for  every  atom  T  except  '_'  and for every term X, (T, X) belongs to Sets (atomic
           sets);

         * (['_'], []) belongs to Sets (the untyped empty set);

         * for every tuple T = {T[1], ..., T[n]} and for every tuple X = {X[1],  ...,  X[n]},  if
           (T[i],  X[i])  belongs  to  Sets  for  every  1  <= i <= n then (T, X) belongs to Sets
           (ordered sets);

         * for every term T, if X is the empty list or a non-empty sorted list [X[1], ...,  X[n]]
           without  duplicates  such  that  (T, X[i]) belongs to Sets for every 1 <= i <= n, then
           ([T], X) belongs to Sets (typed unordered sets).

       An external set is an element of the range of Sets. A type is an element of the domain  of
       Sets.  If S is an element (T, X) of Sets, then T is a valid type of X, T is the type of S,
       and X is the external set of S. from_term/2 creates a set from a type and an  Erlang  term
       turned into an external set.

       The actual sets represented by Sets are the elements of the range of the function Set from
       Sets to Erlang terms and sets of Erlang terms:

         * Set(T,Term) = Term, where T is an atom;

         * Set({T[1], ..., T[n]}, {X[1], ..., X[n]}) = (Set(T[1], X[1]), ..., Set(T[n], X[n]));

         * Set([T], [X[1], ..., X[n]]) = {Set(T, X[1]), ..., Set(T, X[n])};

         * Set([T], []) = {}.

       When there is no risk of confusion, elements of Sets will be identified with the sets they
       represent.  For  instance,  if  U  is  the  result  of  calling  union/2 with S1 and S2 as
       arguments, then U is said to be the union of S1 and S2. A more precise  formulation  would
       be that Set(U) is the union of Set(S1) and Set(S2).

       The  types  are  used to implement the various conditions that sets need to fulfill. As an
       example, consider the relative product of two sets R and S, and recall that  the  relative
       product  of  R  and S is defined if R is a binary relation to Y and S is a binary relation
       from Y. The function that implements the relative product, relative_product/2, checks that
       the arguments represent binary relations by matching [{A,B}] against the type of the first
       argument (Arg1 say), and [{C,D}] against the type of the second argument (Arg2  say).  The
       fact  that  [{A,B}]  matches  the type of Arg1 is to be interpreted as Arg1 representing a
       binary relation from X to Y, where X is defined as all sets Set(x) for some element  x  in
       Sets  the  type of which is A, and similarly for Y. In the same way Arg2 is interpreted as
       representing a binary relation from W to Z. Finally it is checked that B matches C,  which
       is sufficient to ensure that W is equal to Y. The untyped empty set is handled separately:
       its type, ['_'], matches the type of any unordered set.

       A  few  functions  of  this  module  (drestriction/3,  family_projection/2,   partition/2,
       partition_family/2, projection/2, restriction/3, substitution/2) accept an Erlang function
       as a means to modify each element of a given unordered set. Such a function, called SetFun
       in  the following, can be specified as a functional object (fun), a tuple {external, Fun},
       or an integer. If SetFun is specified as a fun, the fun is applied to each element of  the
       given  set  and the return value is assumed to be a set. If SetFun is specified as a tuple
       {external, Fun}, Fun is applied to the external set of each element of the given  set  and
       the  return value is assumed to be an external set. Selecting the elements of an unordered
       set as external sets and assembling a new unordered set from a list of external sets is in
       the  present  implementation more efficient than modifying each element as a set. However,
       this optimization can only be utilized when the elements of the unordered set  are  atomic
       or  ordered  sets.  It  must  also  be the case that the type of the elements matches some
       clause of Fun (the type of the created set is the result of applying Fun to  the  type  of
       the  given set), and that Fun does nothing but selecting, duplicating or rearranging parts
       of the elements. Specifying  a  SetFun  as  an  integer  I  is  equivalent  to  specifying
       {external, fun(X) -> element(I, X) end}, but is to be preferred since it makes it possible
       to handle this case even more efficiently. Examples of SetFuns:

       fun sofs:union/1
       fun(S) -> sofs:partition(1, S) end
       {external, fun(A) -> A end}
       {external, fun({A,_,C}) -> {C,A} end}
       {external, fun({_,{_,C}}) -> C end}
       {external, fun({_,{_,{_,E}=C}}) -> {E,{E,C}} end}
       2

       The order in which a SetFun is applied  to  the  elements  of  an  unordered  set  is  not
       specified, and may change in future versions of sofs.

       The  execution  time  of the functions of this module is dominated by the time it takes to
       sort lists. When  no  sorting  is  needed,  the  execution  time  is  in  the  worst  case
       proportional  to the sum of the sizes of the input arguments and the returned value. A few
       functions execute in constant  time:  from_external,  is_empty_set,  is_set,  is_sofs_set,
       to_external, type.

       The   functions  of  this  module  exit  the  process  with  a  badarg,  bad_function,  or
       type_mismatch message when given badly formed arguments or sets the types of which are not
       compatible.

       When comparing external sets the operator ==/2 is used.

DATA TYPES

       anyset() = ordset() | a_set()

              Any kind of set (also included are the atomic sets).

       binary_relation() = relation()

              A binary relation.

       external_set() = term()

              An external set.

       family() = a_function()

              A family (of subsets).

       a_function() = relation()

              A function.

       ordset()

              An ordered set.

       relation() = a_set()

              An n-ary relation.

       a_set()

              An unordered set.

       set_of_sets() = a_set()

              An unordered set of unordered sets.

       set_fun() = integer() >= 1
                 | {external, fun((external_set()) -> external_set())}
                 | fun((anyset()) -> anyset())

              A SetFun.

       spec_fun() = {external, fun((external_set()) -> boolean())}
                  | fun((anyset()) -> boolean())

       type() = term()

              A type.

       tuple_of(T)

              A tuple where the elements are of type T.

EXPORTS

       a_function(Tuples) -> Function

       a_function(Tuples, Type) -> Function

              Types:

                 Function = a_function()
                 Tuples = [tuple()]
                 Type = type()

              Creates  a  function.  a_function(F,  T)  is  equivalent to from_term(F, T), if the
              result is a function. If no type is explicitly given, [{atom,  atom}]  is  used  as
              type of the function.

       canonical_relation(SetOfSets) -> BinRel

              Types:

                 BinRel = binary_relation()
                 SetOfSets = set_of_sets()

              Returns  the binary relation containing the elements (E, Set) such that Set belongs
              to SetOfSets and E belongs to Set. If SetOfSets is a partition of a set X and R  is
              the  equivalence  relation in X induced by SetOfSets, then the returned relation is
              the canonical map from X onto the equivalence classes with respect to R.

              1> Ss = sofs:from_term([[a,b],[b,c]]),
              CR = sofs:canonical_relation(Ss),
              sofs:to_external(CR).
              [{a,[a,b]},{b,[a,b]},{b,[b,c]},{c,[b,c]}]

       composite(Function1, Function2) -> Function3

              Types:

                 Function1 = Function2 = Function3 = a_function()

              Returns the composite of the functions Function1 and Function2.

              1> F1 = sofs:a_function([{a,1},{b,2},{c,2}]),
              F2 = sofs:a_function([{1,x},{2,y},{3,z}]),
              F = sofs:composite(F1, F2),
              sofs:to_external(F).
              [{a,x},{b,y},{c,y}]

       constant_function(Set, AnySet) -> Function

              Types:

                 AnySet = anyset()
                 Function = a_function()
                 Set = a_set()

              Creates the function that maps each element of the set Set onto AnySet.

              1> S = sofs:set([a,b]),
              E = sofs:from_term(1),
              R = sofs:constant_function(S, E),
              sofs:to_external(R).
              [{a,1},{b,1}]

       converse(BinRel1) -> BinRel2

              Types:

                 BinRel1 = BinRel2 = binary_relation()

              Returns the converse of the binary relation BinRel1.

              1> R1 = sofs:relation([{1,a},{2,b},{3,a}]),
              R2 = sofs:converse(R1),
              sofs:to_external(R2).
              [{a,1},{a,3},{b,2}]

       difference(Set1, Set2) -> Set3

              Types:

                 Set1 = Set2 = Set3 = a_set()

              Returns the difference of the sets Set1 and Set2.

       digraph_to_family(Graph) -> Family

       digraph_to_family(Graph, Type) -> Family

              Types:

                 Graph = digraph()
                 Family = family()
                 Type = type()

              Creates a family from  the  directed  graph  Graph.  Each  vertex  a  of  Graph  is
              represented  by  a  pair  (a,  {b[1],  ...,  b[n]})  where  the b[i]'s are the out-
              neighbours of a. If no type is explicitly given, [{atom, [atom]}] is used  as  type
              of  the  family. It is assumed that Type is a valid type of the external set of the
              family.

              If G is a directed graph, it holds that the vertices and edges of G are the same as
              the vertices and edges of family_to_digraph(digraph_to_family(G)).

       domain(BinRel) -> Set

              Types:

                 BinRel = binary_relation()
                 Set = a_set()

              Returns the domain of the binary relation BinRel.

              1> R = sofs:relation([{1,a},{1,b},{2,b},{2,c}]),
              S = sofs:domain(R),
              sofs:to_external(S).
              [1,2]

       drestriction(BinRel1, Set) -> BinRel2

              Types:

                 BinRel1 = BinRel2 = binary_relation()
                 Set = a_set()

              Returns  the  difference between the binary relation BinRel1 and the restriction of
              BinRel1 to Set.

              1> R1 = sofs:relation([{1,a},{2,b},{3,c}]),
              S = sofs:set([2,4,6]),
              R2 = sofs:drestriction(R1, S),
              sofs:to_external(R2).
              [{1,a},{3,c}]

              drestriction(R, S) is equivalent to difference(R, restriction(R, S)).

       drestriction(SetFun, Set1, Set2) -> Set3

              Types:

                 SetFun = set_fun()
                 Set1 = Set2 = Set3 = a_set()

              Returns a subset of Set1 containing those elements that do not yield an element  in
              Set2 as the result of applying SetFun.

              1> SetFun = {external, fun({_A,B,C}) -> {B,C} end},
              R1 = sofs:relation([{a,aa,1},{b,bb,2},{c,cc,3}]),
              R2 = sofs:relation([{bb,2},{cc,3},{dd,4}]),
              R3 = sofs:drestriction(SetFun, R1, R2),
              sofs:to_external(R3).
              [{a,aa,1}]

              drestriction(F, S1, S2) is equivalent to difference(S1, restriction(F, S1, S2)).

       empty_set() -> Set

              Types:

                 Set = a_set()

              Returns the untyped empty set. empty_set() is equivalent to from_term([], ['_']).

       extension(BinRel1, Set, AnySet) -> BinRel2

              Types:

                 AnySet = anyset()
                 BinRel1 = BinRel2 = binary_relation()
                 Set = a_set()

              Returns  the extension of BinRel1 such that for each element E in Set that does not
              belong to the domain of BinRel1, BinRel2 contains the pair (E, AnySet).

              1> S = sofs:set([b,c]),
              A = sofs:empty_set(),
              R = sofs:family([{a,[1,2]},{b,[3]}]),
              X = sofs:extension(R, S, A),
              sofs:to_external(X).
              [{a,[1,2]},{b,[3]},{c,[]}]

       family(Tuples) -> Family

       family(Tuples, Type) -> Family

              Types:

                 Family = family()
                 Tuples = [tuple()]
                 Type = type()

              Creates a family of subsets. family(F, T) is equivalent to from_term(F, T), if  the
              result  is  a  family.  If no type is explicitly given, [{atom, [atom]}] is used as
              type of the family.

       family_difference(Family1, Family2) -> Family3

              Types:

                 Family1 = Family2 = Family3 = family()

              If Family1 and Family2 are families, then Family3 is the family such that the index
              set  is equal to the index set of Family1, and Family3[i] is the difference between
              Family1[i] and Family2[i] if Family2 maps i, Family1[i] otherwise.

              1> F1 = sofs:family([{a,[1,2]},{b,[3,4]}]),
              F2 = sofs:family([{b,[4,5]},{c,[6,7]}]),
              F3 = sofs:family_difference(F1, F2),
              sofs:to_external(F3).
              [{a,[1,2]},{b,[3]}]

       family_domain(Family1) -> Family2

              Types:

                 Family1 = Family2 = family()

              If Family1 is a family and Family1[i] is a binary relation for every i in the index
              set  of Family1, then Family2 is the family with the same index set as Family1 such
              that Family2[i] is the domain of Family1[i].

              1> FR = sofs:from_term([{a,[{1,a},{2,b},{3,c}]},{b,[]},{c,[{4,d},{5,e}]}]),
              F = sofs:family_domain(FR),
              sofs:to_external(F).
              [{a,[1,2,3]},{b,[]},{c,[4,5]}]

       family_field(Family1) -> Family2

              Types:

                 Family1 = Family2 = family()

              If Family1 is a family and Family1[i] is a binary relation for every i in the index
              set  of Family1, then Family2 is the family with the same index set as Family1 such
              that Family2[i] is the field of Family1[i].

              1> FR = sofs:from_term([{a,[{1,a},{2,b},{3,c}]},{b,[]},{c,[{4,d},{5,e}]}]),
              F = sofs:family_field(FR),
              sofs:to_external(F).
              [{a,[1,2,3,a,b,c]},{b,[]},{c,[4,5,d,e]}]

              family_field(Family1)   is   equivalent   to   family_union(family_domain(Family1),
              family_range(Family1)).

       family_intersection(Family1) -> Family2

              Types:

                 Family1 = Family2 = family()

              If Family1 is a family and Family1[i] is a set of sets for every i in the index set
              of Family1, then Family2 is the family with the same index set as Family1 such that
              Family2[i] is the intersection of Family1[i].

              If  Family1[i]  is  an  empty  set for some i, then the process exits with a badarg
              message.

              1> F1 = sofs:from_term([{a,[[1,2,3],[2,3,4]]},{b,[[x,y,z],[x,y]]}]),
              F2 = sofs:family_intersection(F1),
              sofs:to_external(F2).
              [{a,[2,3]},{b,[x,y]}]

       family_intersection(Family1, Family2) -> Family3

              Types:

                 Family1 = Family2 = Family3 = family()

              If Family1 and Family2 are families, then Family3 is the family such that the index
              set  is  the  intersection of Family1's and Family2's index sets, and Family3[i] is
              the intersection of Family1[i] and Family2[i].

              1> F1 = sofs:family([{a,[1,2]},{b,[3,4]},{c,[5,6]}]),
              F2 = sofs:family([{b,[4,5]},{c,[7,8]},{d,[9,10]}]),
              F3 = sofs:family_intersection(F1, F2),
              sofs:to_external(F3).
              [{b,[4]},{c,[]}]

       family_projection(SetFun, Family1) -> Family2

              Types:

                 SetFun = set_fun()
                 Family1 = Family2 = family()

              If Family1 is a family then Family2 is the  family  with  the  same  index  set  as
              Family1  such  that  Family2[i]  is the result of calling SetFun with Family1[i] as
              argument.

              1> F1 = sofs:from_term([{a,[[1,2],[2,3]]},{b,[[]]}]),
              F2 = sofs:family_projection(fun sofs:union/1, F1),
              sofs:to_external(F2).
              [{a,[1,2,3]},{b,[]}]

       family_range(Family1) -> Family2

              Types:

                 Family1 = Family2 = family()

              If Family1 is a family and Family1[i] is a binary relation for every i in the index
              set  of Family1, then Family2 is the family with the same index set as Family1 such
              that Family2[i] is the range of Family1[i].

              1> FR = sofs:from_term([{a,[{1,a},{2,b},{3,c}]},{b,[]},{c,[{4,d},{5,e}]}]),
              F = sofs:family_range(FR),
              sofs:to_external(F).
              [{a,[a,b,c]},{b,[]},{c,[d,e]}]

       family_specification(Fun, Family1) -> Family2

              Types:

                 Fun = spec_fun()
                 Family1 = Family2 = family()

              If Family1 is a family, then  Family2  is  the  restriction  of  Family1  to  those
              elements  i  of  the index set for which Fun applied to Family1[i] returns true. If
              Fun is a tuple {external, Fun2}, Fun2 is applied to the external set of Family1[i],
              otherwise Fun is applied to Family1[i].

              1> F1 = sofs:family([{a,[1,2,3]},{b,[1,2]},{c,[1]}]),
              SpecFun = fun(S) -> sofs:no_elements(S) =:= 2 end,
              F2 = sofs:family_specification(SpecFun, F1),
              sofs:to_external(F2).
              [{b,[1,2]}]

       family_to_digraph(Family) -> Graph

       family_to_digraph(Family, GraphType) -> Graph

              Types:

                 Graph = digraph()
                 Family = family()
                 GraphType = [digraph:d_type()]

              Creates  a  directed  graph  from  the family Family. For each pair (a, {b[1], ...,
              b[n]}) of Family, the vertex a as well the edges (a, b[i]) for 1  <=  i  <=  n  are
              added to a newly created directed graph.

              If  no  graph type is given  digraph:new/0 is used for creating the directed graph,
              otherwise the GraphType argument is passed on as second argument to digraph:new/1.

              It   F    is    a    family,    it    holds    that    F    is    a    subset    of
              digraph_to_family(family_to_digraph(F),     type(F)).     Equality     holds     if
              union_of_family(F) is a subset of domain(F).

              Creating a cycle in an acyclic graph exits the process with a cyclic message.

       family_to_relation(Family) -> BinRel

              Types:

                 Family = family()
                 BinRel = binary_relation()

              If Family is a family, then BinRel is the binary relation containing all pairs  (i,
              x) such that i belongs to the index set of Family and x belongs to Family[i].

              1> F = sofs:family([{a,[]}, {b,[1]}, {c,[2,3]}]),
              R = sofs:family_to_relation(F),
              sofs:to_external(R).
              [{b,1},{c,2},{c,3}]

       family_union(Family1) -> Family2

              Types:

                 Family1 = Family2 = family()

              If  Family1 is a family and Family1[i] is a set of sets for each i in the index set
              of Family1, then Family2 is the family with the same index set as Family1 such that
              Family2[i] is the union of Family1[i].

              1> F1 = sofs:from_term([{a,[[1,2],[2,3]]},{b,[[]]}]),
              F2 = sofs:family_union(F1),
              sofs:to_external(F2).
              [{a,[1,2,3]},{b,[]}]

              family_union(F) is equivalent to family_projection(fun sofs:union/1, F).

       family_union(Family1, Family2) -> Family3

              Types:

                 Family1 = Family2 = Family3 = family()

              If Family1 and Family2 are families, then Family3 is the family such that the index
              set is the union of Family1's and Family2's index sets, and Family3[i] is the union
              of Family1[i] and Family2[i] if both maps i, Family1[i] or Family2[i] otherwise.

              1> F1 = sofs:family([{a,[1,2]},{b,[3,4]},{c,[5,6]}]),
              F2 = sofs:family([{b,[4,5]},{c,[7,8]},{d,[9,10]}]),
              F3 = sofs:family_union(F1, F2),
              sofs:to_external(F3).
              [{a,[1,2]},{b,[3,4,5]},{c,[5,6,7,8]},{d,[9,10]}]

       field(BinRel) -> Set

              Types:

                 BinRel = binary_relation()
                 Set = a_set()

              Returns the field of the binary relation BinRel.

              1> R = sofs:relation([{1,a},{1,b},{2,b},{2,c}]),
              S = sofs:field(R),
              sofs:to_external(S).
              [1,2,a,b,c]

              field(R) is equivalent to union(domain(R), range(R)).

       from_external(ExternalSet, Type) -> AnySet

              Types:

                 ExternalSet = external_set()
                 AnySet = anyset()
                 Type = type()

              Creates  a  set  from the external set ExternalSet and the type Type. It is assumed
              that Type is a valid type of ExternalSet.

       from_sets(ListOfSets) -> Set

              Types:

                 Set = a_set()
                 ListOfSets = [anyset()]

              Returns the unordered set containing the sets of the list ListOfSets.

              1> S1 = sofs:relation([{a,1},{b,2}]),
              S2 = sofs:relation([{x,3},{y,4}]),
              S = sofs:from_sets([S1,S2]),
              sofs:to_external(S).
              [[{a,1},{b,2}],[{x,3},{y,4}]]

       from_sets(TupleOfSets) -> Ordset

              Types:

                 Ordset = ordset()
                 TupleOfSets = tuple_of(anyset())

              Returns the ordered set containing the sets of the non-empty tuple TupleOfSets.

       from_term(Term) -> AnySet

       from_term(Term, Type) -> AnySet

              Types:

                 AnySet = anyset()
                 Term = term()
                 Type = type()

              Creates an element of Sets by traversing the term  Term,  sorting  lists,  removing
              duplicates and deriving or verifying a valid type for the so obtained external set.
              An explicitly given type Type can be used to limit the depth of the  traversal;  an
              atomic  type  stops  the traversal, as demonstrated by this example where "foo" and
              {"foo"} are left unmodified:

              1> S = sofs:from_term([{{"foo"},[1,1]},{"foo",[2,2]}], [{atom,[atom]}]),
              sofs:to_external(S).
              [{{"foo"},[1]},{"foo",[2]}]

              from_term can be used for creating atomic or ordered sets. The only purpose of such
              a  set  is that of later building unordered sets since all functions in this module
              that do anything  operate  on  unordered  sets.  Creating  unordered  sets  from  a
              collection of ordered sets may be the way to go if the ordered sets are big and one
              does not want to waste heap by rebuilding the elements of  the  unordered  set.  An
              example showing that a set can be built "layer by layer":

              1> A = sofs:from_term(a),
              S = sofs:set([1,2,3]),
              P1 = sofs:from_sets({A,S}),
              P2 = sofs:from_term({b,[6,5,4]}),
              Ss = sofs:from_sets([P1,P2]),
              sofs:to_external(Ss).
              [{a,[1,2,3]},{b,[4,5,6]}]

              Other functions that create sets are from_external/2 and from_sets/1. Special cases
              of from_term/2  are  a_function/1,2,  empty_set/0,  family/1,2,  relation/1,2,  and
              set/1,2.

       image(BinRel, Set1) -> Set2

              Types:

                 BinRel = binary_relation()
                 Set1 = Set2 = a_set()

              Returns the image of the set Set1 under the binary relation BinRel.

              1> R = sofs:relation([{1,a},{2,b},{2,c},{3,d}]),
              S1 = sofs:set([1,2]),
              S2 = sofs:image(R, S1),
              sofs:to_external(S2).
              [a,b,c]

       intersection(SetOfSets) -> Set

              Types:

                 Set = a_set()
                 SetOfSets = set_of_sets()

              Returns the intersection of the set of sets SetOfSets.

              Intersecting an empty set of sets exits the process with a badarg message.

       intersection(Set1, Set2) -> Set3

              Types:

                 Set1 = Set2 = Set3 = a_set()

              Returns the intersection of Set1 and Set2.

       intersection_of_family(Family) -> Set

              Types:

                 Family = family()
                 Set = a_set()

              Returns the intersection of the family Family.

              Intersecting an empty family exits the process with a badarg message.

              1> F = sofs:family([{a,[0,2,4]},{b,[0,1,2]},{c,[2,3]}]),
              S = sofs:intersection_of_family(F),
              sofs:to_external(S).
              [2]

       inverse(Function1) -> Function2

              Types:

                 Function1 = Function2 = a_function()

              Returns the inverse of the function Function1.

              1> R1 = sofs:relation([{1,a},{2,b},{3,c}]),
              R2 = sofs:inverse(R1),
              sofs:to_external(R2).
              [{a,1},{b,2},{c,3}]

       inverse_image(BinRel, Set1) -> Set2

              Types:

                 BinRel = binary_relation()
                 Set1 = Set2 = a_set()

              Returns the inverse image of Set1 under the binary relation BinRel.

              1> R = sofs:relation([{1,a},{2,b},{2,c},{3,d}]),
              S1 = sofs:set([c,d,e]),
              S2 = sofs:inverse_image(R, S1),
              sofs:to_external(S2).
              [2,3]

       is_a_function(BinRel) -> Bool

              Types:

                 Bool = boolean()
                 BinRel = binary_relation()

              Returns  true if the binary relation BinRel is a function or the untyped empty set,
              false otherwise.

       is_disjoint(Set1, Set2) -> Bool

              Types:

                 Bool = boolean()
                 Set1 = Set2 = a_set()

              Returns true if Set1 and Set2 are disjoint, false otherwise.

       is_empty_set(AnySet) -> Bool

              Types:

                 AnySet = anyset()
                 Bool = boolean()

              Returns true if AnySet is an empty unordered set, false otherwise.

       is_equal(AnySet1, AnySet2) -> Bool

              Types:

                 AnySet1 = AnySet2 = anyset()
                 Bool = boolean()

              Returns true if the AnySet1 and AnySet2 are equal, false  otherwise.  This  example
              shows that ==/2 is used when comparing sets for equality:

              1> S1 = sofs:set([1.0]),
              S2 = sofs:set([1]),
              sofs:is_equal(S1, S2).
              true

       is_set(AnySet) -> Bool

              Types:

                 AnySet = anyset()
                 Bool = boolean()

              Returns  true  if AnySet is an unordered set, and false if AnySet is an ordered set
              or an atomic set.

       is_sofs_set(Term) -> Bool

              Types:

                 Bool = boolean()
                 Term = term()

              Returns true if Term is an unordered set, an ordered set or an  atomic  set,  false
              otherwise.

       is_subset(Set1, Set2) -> Bool

              Types:

                 Bool = boolean()
                 Set1 = Set2 = a_set()

              Returns true if Set1 is a subset of Set2, false otherwise.

       is_type(Term) -> Bool

              Types:

                 Bool = boolean()
                 Term = term()

              Returns true if the term Term is a type.

       join(Relation1, I, Relation2, J) -> Relation3

              Types:

                 Relation1 = Relation2 = Relation3 = relation()
                 I = J = integer() >= 1

              Returns  the natural join of the relations Relation1 and Relation2 on coordinates I
              and J.

              1> R1 = sofs:relation([{a,x,1},{b,y,2}]),
              R2 = sofs:relation([{1,f,g},{1,h,i},{2,3,4}]),
              J = sofs:join(R1, 3, R2, 1),
              sofs:to_external(J).
              [{a,x,1,f,g},{a,x,1,h,i},{b,y,2,3,4}]

       multiple_relative_product(TupleOfBinRels, BinRel1) -> BinRel2

              Types:

                 TupleOfBinRels = tuple_of(BinRel)
                 BinRel = BinRel1 = BinRel2 = binary_relation()

              If TupleOfBinRels is a non-empty tuple {R[1], ..., R[n]} of  binary  relations  and
              BinRel1  is a binary relation, then BinRel2 is the multiple relative product of the
              ordered set (R[i], ..., R[n]) and BinRel1.

              1> Ri = sofs:relation([{a,1},{b,2},{c,3}]),
              R = sofs:relation([{a,b},{b,c},{c,a}]),
              MP = sofs:multiple_relative_product({Ri, Ri}, R),
              sofs:to_external(sofs:range(MP)).
              [{1,2},{2,3},{3,1}]

       no_elements(ASet) -> NoElements

              Types:

                 ASet = a_set() | ordset()
                 NoElements = integer() >= 0

              Returns the number of elements of the ordered or unordered set ASet.

       partition(SetOfSets) -> Partition

              Types:

                 SetOfSets = set_of_sets()
                 Partition = a_set()

              Returns the partition of the union of the set  of  sets  SetOfSets  such  that  two
              elements are considered equal if they belong to the same elements of SetOfSets.

              1> Sets1 = sofs:from_term([[a,b,c],[d,e,f],[g,h,i]]),
              Sets2 = sofs:from_term([[b,c,d],[e,f,g],[h,i,j]]),
              P = sofs:partition(sofs:union(Sets1, Sets2)),
              sofs:to_external(P).
              [[a],[b,c],[d],[e,f],[g],[h,i],[j]]

       partition(SetFun, Set) -> Partition

              Types:

                 SetFun = set_fun()
                 Partition = Set = a_set()

              Returns  the  partition  of  Set such that two elements are considered equal if the
              results of applying SetFun are equal.

              1> Ss = sofs:from_term([[a],[b],[c,d],[e,f]]),
              SetFun = fun(S) -> sofs:from_term(sofs:no_elements(S)) end,
              P = sofs:partition(SetFun, Ss),
              sofs:to_external(P).
              [[[a],[b]],[[c,d],[e,f]]]

       partition(SetFun, Set1, Set2) -> {Set3, Set4}

              Types:

                 SetFun = set_fun()
                 Set1 = Set2 = Set3 = Set4 = a_set()

              Returns a pair of sets that, regarded as constituting a set, forms a  partition  of
              Set1.  If  the result of applying SetFun to an element of Set1 yields an element in
              Set2, the element belongs to Set3, otherwise the element belongs to Set4.

              1> R1 = sofs:relation([{1,a},{2,b},{3,c}]),
              S = sofs:set([2,4,6]),
              {R2,R3} = sofs:partition(1, R1, S),
              {sofs:to_external(R2),sofs:to_external(R3)}.
              {[{2,b}],[{1,a},{3,c}]}

              partition(F, S1, S2) is equivalent to {restriction(F, S1, S2), drestriction(F,  S1,
              S2)}.

       partition_family(SetFun, Set) -> Family

              Types:

                 Family = family()
                 SetFun = set_fun()
                 Set = a_set()

              Returns the family Family where the indexed set is a partition of Set such that two
              elements are considered equal if the results of applying SetFun are the same  value
              i. This i is the index that Family maps onto the equivalence class.

              1> S = sofs:relation([{a,a,a,a},{a,a,b,b},{a,b,b,b}]),
              SetFun = {external, fun({A,_,C,_}) -> {A,C} end},
              F = sofs:partition_family(SetFun, S),
              sofs:to_external(F).
              [{{a,a},[{a,a,a,a}]},{{a,b},[{a,a,b,b},{a,b,b,b}]}]

       product(TupleOfSets) -> Relation

              Types:

                 Relation = relation()
                 TupleOfSets = tuple_of(a_set())

              Returns the Cartesian product of the non-empty tuple of sets TupleOfSets. If (x[1],
              ..., x[n]) is an element of the n-ary relation Relation, then x[i]  is  drawn  from
              element i of TupleOfSets.

              1> S1 = sofs:set([a,b]),
              S2 = sofs:set([1,2]),
              S3 = sofs:set([x,y]),
              P3 = sofs:product({S1,S2,S3}),
              sofs:to_external(P3).
              [{a,1,x},{a,1,y},{a,2,x},{a,2,y},{b,1,x},{b,1,y},{b,2,x},{b,2,y}]

       product(Set1, Set2) -> BinRel

              Types:

                 BinRel = binary_relation()
                 Set1 = Set2 = a_set()

              Returns the Cartesian product of Set1 and Set2.

              1> S1 = sofs:set([1,2]),
              S2 = sofs:set([a,b]),
              R = sofs:product(S1, S2),
              sofs:to_external(R).
              [{1,a},{1,b},{2,a},{2,b}]

              product(S1, S2) is equivalent to product({S1, S2}).

       projection(SetFun, Set1) -> Set2

              Types:

                 SetFun = set_fun()
                 Set1 = Set2 = a_set()

              Returns  the  set  created  by  substituting  each element of Set1 by the result of
              applying SetFun to the element.

              If SetFun is a number i >= 1 and Set1 is a relation, then the returned set  is  the
              projection of Set1 onto coordinate i.

              1> S1 = sofs:from_term([{1,a},{2,b},{3,a}]),
              S2 = sofs:projection(2, S1),
              sofs:to_external(S2).
              [a,b]

       range(BinRel) -> Set

              Types:

                 BinRel = binary_relation()
                 Set = a_set()

              Returns the range of the binary relation BinRel.

              1> R = sofs:relation([{1,a},{1,b},{2,b},{2,c}]),
              S = sofs:range(R),
              sofs:to_external(S).
              [a,b,c]

       relation(Tuples) -> Relation

       relation(Tuples, Type) -> Relation

              Types:

                 N = integer()
                 Type = N | type()
                 Relation = relation()
                 Tuples = [tuple()]

              Creates a relation. relation(R, T) is equivalent to from_term(R, T), if T is a type
              and the result is a relation. If Type is an integer N, then [{atom,  ...,  atom}]),
              where  the  size  of the tuple is N, is used as type of the relation. If no type is
              explicitly given, the size of the first tuple of Tuples is used if there is such  a
              tuple. relation([]) is equivalent to relation([], 2).

       relation_to_family(BinRel) -> Family

              Types:

                 Family = family()
                 BinRel = binary_relation()

              Returns  the  family  Family  such that the index set is equal to the domain of the
              binary relation BinRel, and Family[i] is the image of the set of i under BinRel.

              1> R = sofs:relation([{b,1},{c,2},{c,3}]),
              F = sofs:relation_to_family(R),
              sofs:to_external(F).
              [{b,[1]},{c,[2,3]}]

       relative_product(ListOfBinRels) -> BinRel2

       relative_product(ListOfBinRels, BinRel1) -> BinRel2

              Types:

                 ListOfBinRels = [BinRel, ...]
                 BinRel = BinRel1 = BinRel2 = binary_relation()

              If ListOfBinRels is a non-empty list [R[1], ...,  R[n]]  of  binary  relations  and
              BinRel1  is  a binary relation, then BinRel2 is the relative product of the ordered
              set (R[i], ..., R[n]) and BinRel1.

              If BinRel1 is omitted, the  relation  of  equality  between  the  elements  of  the
              Cartesian  product  of  the  ranges of R[i], range R[1] x ... x range R[n], is used
              instead (intuitively, nothing is "lost").

              1> TR = sofs:relation([{1,a},{1,aa},{2,b}]),
              R1 = sofs:relation([{1,u},{2,v},{3,c}]),
              R2 = sofs:relative_product([TR, R1]),
              sofs:to_external(R2).
              [{1,{a,u}},{1,{aa,u}},{2,{b,v}}]

              Note that relative_product([R1], R2) is different  from  relative_product(R1,  R2);
              the list of one element is not identified with the element itself.

       relative_product(BinRel1, BinRel2) -> BinRel3

              Types:

                 BinRel1 = BinRel2 = BinRel3 = binary_relation()

              Returns the relative product of the binary relations BinRel1 and BinRel2.

       relative_product1(BinRel1, BinRel2) -> BinRel3

              Types:

                 BinRel1 = BinRel2 = BinRel3 = binary_relation()

              Returns the relative product of the converse of the binary relation BinRel1 and the
              binary relation BinRel2.

              1> R1 = sofs:relation([{1,a},{1,aa},{2,b}]),
              R2 = sofs:relation([{1,u},{2,v},{3,c}]),
              R3 = sofs:relative_product1(R1, R2),
              sofs:to_external(R3).
              [{a,u},{aa,u},{b,v}]

              relative_product1(R1, R2) is equivalent to relative_product(converse(R1), R2).

       restriction(BinRel1, Set) -> BinRel2

              Types:

                 BinRel1 = BinRel2 = binary_relation()
                 Set = a_set()

              Returns the restriction of the binary relation BinRel1 to Set.

              1> R1 = sofs:relation([{1,a},{2,b},{3,c}]),
              S = sofs:set([1,2,4]),
              R2 = sofs:restriction(R1, S),
              sofs:to_external(R2).
              [{1,a},{2,b}]

       restriction(SetFun, Set1, Set2) -> Set3

              Types:

                 SetFun = set_fun()
                 Set1 = Set2 = Set3 = a_set()

              Returns a subset of Set1 containing those elements that yield an element in Set2 as
              the result of applying SetFun.

              1> S1 = sofs:relation([{1,a},{2,b},{3,c}]),
              S2 = sofs:set([b,c,d]),
              S3 = sofs:restriction(2, S1, S2),
              sofs:to_external(S3).
              [{2,b},{3,c}]

       set(Terms) -> Set

       set(Terms, Type) -> Set

              Types:

                 Set = a_set()
                 Terms = [term()]
                 Type = type()

              Creates an unordered set. set(L, T) is equivalent to from_term(L, T), if the result
              is an unordered set. If no type is explicitly given, [atom] is used as type of  the
              set.

       specification(Fun, Set1) -> Set2

              Types:

                 Fun = spec_fun()
                 Set1 = Set2 = a_set()

              Returns the set containing every element of Set1 for which Fun returns true. If Fun
              is a tuple {external, Fun2}, Fun2 is applied to the external set of  each  element,
              otherwise Fun is applied to each element.

              1> R1 = sofs:relation([{a,1},{b,2}]),
              R2 = sofs:relation([{x,1},{x,2},{y,3}]),
              S1 = sofs:from_sets([R1,R2]),
              S2 = sofs:specification(fun sofs:is_a_function/1, S1),
              sofs:to_external(S2).
              [[{a,1},{b,2}]]

       strict_relation(BinRel1) -> BinRel2

              Types:

                 BinRel1 = BinRel2 = binary_relation()

              Returns the strict relation corresponding to the binary relation BinRel1.

              1> R1 = sofs:relation([{1,1},{1,2},{2,1},{2,2}]),
              R2 = sofs:strict_relation(R1),
              sofs:to_external(R2).
              [{1,2},{2,1}]

       substitution(SetFun, Set1) -> Set2

              Types:

                 SetFun = set_fun()
                 Set1 = Set2 = a_set()

              Returns  a  function,  the  domain of which is Set1. The value of an element of the
              domain is the result of applying SetFun to the element.

              1> L = [{a,1},{b,2}].
              [{a,1},{b,2}]
              2> sofs:to_external(sofs:projection(1,sofs:relation(L))).
              [a,b]
              3> sofs:to_external(sofs:substitution(1,sofs:relation(L))).
              [{{a,1},a},{{b,2},b}]
              4> SetFun = {external, fun({A,_}=E) -> {E,A} end},
              sofs:to_external(sofs:projection(SetFun,sofs:relation(L))).
              [{{a,1},a},{{b,2},b}]

              The relation of equality between the elements of {a,b,c}:

              1> I = sofs:substitution(fun(A) -> A end, sofs:set([a,b,c])),
              sofs:to_external(I).
              [{a,a},{b,b},{c,c}]

              Let SetOfSets be a set of sets and BinRel a binary relation. The function that maps
              each  element  Set  of  SetOfSets onto the image of Set under BinRel is returned by
              this function:

              images(SetOfSets, BinRel) ->
                 Fun = fun(Set) -> sofs:image(BinRel, Set) end,
                 sofs:substitution(Fun, SetOfSets).

              Here might be the place to reveal something that was more or  less  stated  before,
              namely  that  external  unordered  sets  are  represented  as  sorted  lists.  As a
              consequence, creating the image of a set  under  a  relation  R  may  traverse  all
              elements  of  R  (to  that  comes  the sorting of results, the image). In images/2,
              BinRel will be traversed once for each element of SetOfSets,  which  may  take  too
              long.  The  following efficient function could be used instead under the assumption
              that the image of each element of SetOfSets under BinRel is non-empty:

              images2(SetOfSets, BinRel) ->
                 CR = sofs:canonical_relation(SetOfSets),
                 R = sofs:relative_product1(CR, BinRel),
                 sofs:relation_to_family(R).

       symdiff(Set1, Set2) -> Set3

              Types:

                 Set1 = Set2 = Set3 = a_set()

              Returns the symmetric difference (or the Boolean sum) of Set1 and Set2.

              1> S1 = sofs:set([1,2,3]),
              S2 = sofs:set([2,3,4]),
              P = sofs:symdiff(S1, S2),
              sofs:to_external(P).
              [1,4]

       symmetric_partition(Set1, Set2) -> {Set3, Set4, Set5}

              Types:

                 Set1 = Set2 = Set3 = Set4 = Set5 = a_set()

              Returns a triple of sets: Set3 contains the elements of Set1 that do not belong  to
              Set2;  Set4  contains  the  elements of Set1 that belong to Set2; Set5 contains the
              elements of Set2 that do not belong to Set1.

       to_external(AnySet) -> ExternalSet

              Types:

                 ExternalSet = external_set()
                 AnySet = anyset()

              Returns the external set of an atomic, ordered or unordered set.

       to_sets(ASet) -> Sets

              Types:

                 ASet = a_set() | ordset()
                 Sets = tuple_of(AnySet) | [AnySet]
                 AnySet = anyset()

              Returns the elements of the ordered set ASet as a tuple of sets, and  the  elements
              of the unordered set ASet as a sorted list of sets without duplicates.

       type(AnySet) -> Type

              Types:

                 AnySet = anyset()
                 Type = type()

              Returns the type of an atomic, ordered or unordered set.

       union(SetOfSets) -> Set

              Types:

                 Set = a_set()
                 SetOfSets = set_of_sets()

              Returns the union of the set of sets SetOfSets.

       union(Set1, Set2) -> Set3

              Types:

                 Set1 = Set2 = Set3 = a_set()

              Returns the union of Set1 and Set2.

       union_of_family(Family) -> Set

              Types:

                 Family = family()
                 Set = a_set()

              Returns the union of the family Family.

              1> F = sofs:family([{a,[0,2,4]},{b,[0,1,2]},{c,[2,3]}]),
              S = sofs:union_of_family(F),
              sofs:to_external(S).
              [0,1,2,3,4]

       weak_relation(BinRel1) -> BinRel2

              Types:

                 BinRel1 = BinRel2 = binary_relation()

              Returns  a  subset  S  of  the weak relation W corresponding to the binary relation
              BinRel1. Let F be the field of BinRel1. The subset S is defined so that x S y if  x
              W y for some x in F and for some y in F.

              1> R1 = sofs:relation([{1,1},{1,2},{3,1}]),
              R2 = sofs:weak_relation(R1),
              sofs:to_external(R2).
              [{1,1},{1,2},{2,2},{3,1},{3,3}]

SEE ALSO

       dict(3erl), digraph(3erl), orddict(3erl), ordsets(3erl), sets(3erl)