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**NAME**

sofs - Functions for Manipulating Sets of Sets

**DESCRIPTION**

Thesofsmodule 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)}. Theunorderedsetcontaining the elements a, b and c is denoted {a, b, c}. This notation is not to be confused with tuples. Theorderedpairof a and b, with firstcoordinatea and second coordinate b, is denoted (a, b). An ordered pair is anorderedsetof 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 theemptyset. If two sets A and B contain the same elements, then A isequalto 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 asubsetof A. Theunionof two sets A and B is the smallest set that contains all elements of A and all elements of B. Theintersectionof two sets A and B is the set that contains all elements of A that belong to B. Two sets aredisjointif their intersection is the empty set. Thedifferenceof two sets A and B is the set that contains all elements of A that do not belong to B. Thesymmetricdifferenceof two sets is the set that contains those element that belong to either of the two sets, but not both. Theunionof a collection of sets is the smallest set that contains all the elements that belong to at least one set of the collection. Theintersectionof a non-empty collection of sets is the set that contains all elements that belong to every set of the collection. TheCartesianproductof 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}. Arelationis 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. Thedomainof R is the set {x : x R y for some y in Y}. Therangeof R is the set {y : x R y for some x in X}. Theconverseof R is the set {a : a = (y, x) for some (x, y) in R}. If A is a subset of X, then theimageof A under R is the set {y : x R y for some x in A}, and if B is a subset of Y, then theinverseimageof 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 therelativeproductof 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. Therestrictionof 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 anextensionof S to X. If X = Y then we call R a relationinX. Thefieldof 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 thestrictrelation 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 theweakrelation corresponding to S. A relation R in X isreflexiveif x R x for every element x of X; it issymmetricif x R y implies that y R x; and it istransitiveif x R y and y R z imply that x R z. AfunctionF 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 theinverseof F. The relative product of two functions F1 and F2 is called thecompositeof 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 afamily. The domain of a family is called theindexset, and the range is called theindexedset. 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 afamilyofsubsetsof X. If x is a family of subsets of X, then the union of the range of x is called theunionofthefamilyx. If x is non-empty (the index set is non-empty), theintersectionofthefamilyx 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. Apartitionof 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 anequivalencerelationif it is reflexive, symmetric and transitive. If R is an equivalence relation in X, and x is an element of X, theequivalenceclassof 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 thecanonicalmapis 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 asbinaryrelations. 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. Theprojectionof 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. Therelativeproductof 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]. Themultiplerelativeproductof 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}. Thenaturaljoinof 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 (atomicsets); * (['_'], []) belongs to Sets (theuntypedemptyset); * 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 (orderedsets); * 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 (typedunorderedsets). Anexternalsetis an element of the range of Sets. Atypeis an element of the domain of Sets. If S is an element (T, X) of Sets, then T is avalidtypeof X, T is the type of S, and X is the external set of S.from_term/2creates 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 callingunion/2with 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 abadarg,bad_function, ortype_mismatchmessage when given badly formed arguments or sets the types of which are not compatible. When comparing external sets the operator==/2is used.

**DATA** **TYPES**

anyset()=ordset()|a_set()Any kind of set (also included are the atomic sets).binary_relation()=relation()Abinaryrelation.external_set()= term() Anexternalset.family()=a_function()Afamily(of subsets).a_function()=relation()Afunction.ordset()Anorderedset.relation()=a_set()Ann-aryrelation.a_set()Anunorderedset.set_of_sets()=a_set()Anunorderedsetof unordered sets.set_fun()= integer() >= 1 | {external, fun((external_set()) ->external_set())} | fun((anyset()) ->anyset()) ASetFun.spec_fun()= {external, fun((external_set()) -> boolean())} | fun((anyset()) -> boolean())type()= term() Atype.tuple_of(T)A tuple where the elements are of typeT.

**EXPORTS**

a_function(Tuples)->Functiona_function(Tuples,Type)->FunctionTypes: Function =a_function()Tuples = [tuple()] Type =type()Creates afunction.a_function(F,T)is equivalent tofrom_term(F,T), if the result is a function. If notypeis explicitly given,[{atom,atom}]is used as type of the function.canonical_relation(SetOfSets)->BinRelTypes: 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 apartitionof a set X and R is the equivalence relation in X induced by SetOfSets, then the returned relation is thecanonicalmapfrom 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)->Function3Types: Function1 = Function2 = Function3 =a_function()Returns thecompositeof 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)->FunctionTypes: AnySet =anyset()Function =a_function()Set =a_set()Creates thefunctionthat 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)->BinRel2Types: BinRel1 = BinRel2 =binary_relation()Returns theconverseof 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)->Set3Types: Set1 = Set2 = Set3 =a_set()Returns thedifferenceof the sets Set1 and Set2.digraph_to_family(Graph)->Familydigraph_to_family(Graph,Type)->FamilyTypes: Graph = digraph() Family =family()Type =type()Creates afamilyfrom 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 avalidtypeof 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 offamily_to_digraph(digraph_to_family(G)).domain(BinRel)->SetTypes: BinRel =binary_relation()Set =a_set()Returns thedomainof 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)->BinRel2Types: BinRel1 = BinRel2 =binary_relation()Set =a_set()Returns the difference between the binary relation BinRel1 and therestrictionof 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 todifference(R,restriction(R,S)).drestriction(SetFun,Set1,Set2)->Set3Types: 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 todifference(S1,restriction(F,S1,S2)).empty_set()->SetTypes: Set =a_set()Returns theuntypedemptyset.empty_set()is equivalent tofrom_term([],['_']).extension(BinRel1,Set,AnySet)->BinRel2Types: AnySet =anyset()BinRel1 = BinRel2 =binary_relation()Set =a_set()Returns theextensionof BinRel1 such that for each element E in Set that does not belong to thedomainof 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)->Familyfamily(Tuples,Type)->FamilyTypes: Family =family()Tuples = [tuple()] Type =type()Creates afamilyofsubsets.family(F,T)is equivalent tofrom_term(F,T), if the result is a family. If notypeis explicitly given,[{atom,[atom]}]is used as type of the family.family_difference(Family1,Family2)->Family3Types: Family1 = Family2 = Family3 =family()If Family1 and Family2 arefamilies, 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)->Family2Types: Family1 = Family2 =family()If Family1 is afamilyand 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 thedomainof 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)->Family2Types: Family1 = Family2 =family()If Family1 is afamilyand 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 thefieldof 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 tofamily_union(family_domain(Family1),family_range(Family1)).family_intersection(Family1)->Family2Types: Family1 = Family2 =family()If Family1 is afamilyand 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 theintersectionof Family1[i]. If Family1[i] is an empty set for some i, then the process exits with abadargmessage. 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)->Family3Types: Family1 = Family2 = Family3 =family()If Family1 and Family2 arefamilies, 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)->Family2Types: SetFun =set_fun()Family1 = Family2 =family()If Family1 is afamilythen 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)->Family2Types: Family1 = Family2 =family()If Family1 is afamilyand 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 therangeof 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)->Family2Types: Fun =spec_fun()Family1 = Family2 =family()If Family1 is afamily, then Family2 is therestrictionof Family1 to those elements i of the index set for which Fun applied to Family1[i] returnstrue. If Fun is a tuple{external,Fun2}, Fun2 is applied to theexternalsetof 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)->Graphfamily_to_digraph(Family,GraphType)->GraphTypes: Graph = digraph() Family =family()GraphType = [digraph:d_type()] Creates a directed graph from thefamilyFamily. 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 givendigraph:new/0is used for creating the directed graph, otherwise the GraphType argument is passed on as second argument todigraph:new/1. It F is a family, it holds that F is a subset ofdigraph_to_family(family_to_digraph(F),type(F)). Equality holds ifunion_of_family(F)is a subset ofdomain(F). Creating a cycle in an acyclic graph exits the process with acyclicmessage.family_to_relation(Family)->BinRelTypes: Family =family()BinRel =binary_relation()If Family is afamily, 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)->Family2Types: Family1 = Family2 =family()If Family1 is afamilyand 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 theunionof 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 tofamily_projection(funsofs:union/1,F).family_union(Family1,Family2)->Family3Types: Family1 = Family2 = Family3 =family()If Family1 and Family2 arefamilies, 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)->SetTypes: BinRel =binary_relation()Set =a_set()Returns thefieldof 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 tounion(domain(R),range(R)).from_external(ExternalSet,Type)->AnySetTypes: ExternalSet =external_set()AnySet =anyset()Type =type()Creates a set from theexternalsetExternalSet and thetypeType. It is assumed that Type is avalidtypeof ExternalSet.from_sets(ListOfSets)->SetTypes: Set =a_set()ListOfSets = [anyset()] Returns theunorderedsetcontaining 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)->OrdsetTypes: Ordset =ordset()TupleOfSets =tuple_of(anyset()) Returns theorderedsetcontaining the sets of the non-empty tuple TupleOfSets.from_term(Term)->AnySetfrom_term(Term,Type)->AnySetTypes: AnySet =anyset()Term = term() Type =type()Creates an element ofSetsby traversing the term Term, sorting lists, removing duplicates and deriving or verifying avalidtypefor the so obtained external set. An explicitly giventypeType 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_termcan 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 thatdoanything 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 arefrom_external/2andfrom_sets/1. Special cases offrom_term/2area_function/1,2,empty_set/0,family/1,2,relation/1,2, andset/1,2.image(BinRel,Set1)->Set2Types: BinRel =binary_relation()Set1 = Set2 =a_set()Returns theimageof 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)->SetTypes: Set =a_set()SetOfSets =set_of_sets()Returns theintersectionof the set of sets SetOfSets. Intersecting an empty set of sets exits the process with abadargmessage.intersection(Set1,Set2)->Set3Types: Set1 = Set2 = Set3 =a_set()Returns theintersectionof Set1 and Set2.intersection_of_family(Family)->SetTypes: Family =family()Set =a_set()Returns the intersection of thefamilyFamily. Intersecting an empty family exits the process with abadargmessage. 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)->Function2Types: Function1 = Function2 =a_function()Returns theinverseof 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)->Set2Types: BinRel =binary_relation()Set1 = Set2 =a_set()Returns theinverseimageof 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)->BoolTypes: Bool = boolean() BinRel =binary_relation()Returnstrueif the binary relation BinRel is afunctionor the untyped empty set,falseotherwise.is_disjoint(Set1,Set2)->BoolTypes: Bool = boolean() Set1 = Set2 =a_set()Returnstrueif Set1 and Set2 aredisjoint,falseotherwise.is_empty_set(AnySet)->BoolTypes: AnySet =anyset()Bool = boolean() Returnstrueif AnySet is an empty unordered set,falseotherwise.is_equal(AnySet1,AnySet2)->BoolTypes: AnySet1 = AnySet2 =anyset()Bool = boolean() Returnstrueif the AnySet1 and AnySet2 areequal,falseotherwise. This example shows that==/2is used when comparing sets for equality: 1> S1 = sofs:set([1.0]), S2 = sofs:set([1]), sofs:is_equal(S1, S2). trueis_set(AnySet)->BoolTypes: AnySet =anyset()Bool = boolean() Returnstrueif AnySet is anunorderedset, andfalseif AnySet is an ordered set or an atomic set.is_sofs_set(Term)->BoolTypes: Bool = boolean() Term = term() Returnstrueif Term is anunorderedset, an ordered set or an atomic set,falseotherwise.is_subset(Set1,Set2)->BoolTypes: Bool = boolean() Set1 = Set2 =a_set()Returnstrueif Set1 is asubsetof Set2,falseotherwise.is_type(Term)->BoolTypes: Bool = boolean() Term = term() Returnstrueif the term Term is atype.join(Relation1,I,Relation2,J)->Relation3Types: Relation1 = Relation2 = Relation3 =relation()I = J = integer() >= 1 Returns thenaturaljoinof 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)->BinRel2Types: 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 themultiplerelativeproductof 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)->NoElementsTypes: ASet =a_set()|ordset()NoElements = integer() >= 0 Returns the number of elements of the ordered or unordered set ASet.partition(SetOfSets)->PartitionTypes: SetOfSets =set_of_sets()Partition =a_set()Returns thepartitionof 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)->PartitionTypes: SetFun =set_fun()Partition = Set =a_set()Returns thepartitionof 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 apartitionof 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)->FamilyTypes: Family =family()SetFun =set_fun()Set =a_set()Returns thefamilyFamily where the indexed set is apartitionof 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 theequivalenceclass. 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)->RelationTypes: Relation =relation()TupleOfSets =tuple_of(a_set()) Returns theCartesianproductof 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)->BinRelTypes: BinRel =binary_relation()Set1 = Set2 =a_set()Returns theCartesianproductof 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 toproduct({S1,S2}).projection(SetFun,Set1)->Set2Types: 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 theprojectionof 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)->SetTypes: BinRel =binary_relation()Set =a_set()Returns therangeof 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)->Relationrelation(Tuples,Type)->RelationTypes: N = integer() Type = N |type()Relation =relation()Tuples = [tuple()] Creates arelation.relation(R,T)is equivalent tofrom_term(R,T), if T is atypeand 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 torelation([],2).relation_to_family(BinRel)->FamilyTypes: Family =family()BinRel =binary_relation()Returns thefamilyFamily such that the index set is equal to thedomainof the binary relation BinRel, and Family[i] is theimageof 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)->BinRel2relative_product(ListOfBinRels,BinRel1)->BinRel2Types: 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 therelativeproductof the ordered set (R[i], ..., R[n]) and BinRel1. If BinRel1 is omitted, the relation of equality between the elements of theCartesianproductof 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 thatrelative_product([R1],R2)is different fromrelative_product(R1,R2); the list of one element is not identified with the element itself.relative_product(BinRel1,BinRel2)->BinRel3Types: BinRel1 = BinRel2 = BinRel3 =binary_relation()Returns therelativeproductof the binary relations BinRel1 and BinRel2.relative_product1(BinRel1,BinRel2)->BinRel3Types: BinRel1 = BinRel2 = BinRel3 =binary_relation()Returns therelativeproductof theconverseof 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 torelative_product(converse(R1),R2).restriction(BinRel1,Set)->BinRel2Types: BinRel1 = BinRel2 =binary_relation()Set =a_set()Returns therestrictionof 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)->Set3Types: 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)->Setset(Terms,Type)->SetTypes: Set =a_set()Terms = [term()] Type =type()Creates anunorderedset.set(L,T)is equivalent tofrom_term(L,T), if the result is an unordered set. If notypeis explicitly given,[atom]is used as type of the set.specification(Fun,Set1)->Set2Types: Fun =spec_fun()Set1 = Set2 =a_set()Returns the set containing every element of Set1 for which Fun returnstrue. If Fun is a tuple{external,Fun2}, Fun2 is applied to theexternalsetof 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)->BinRel2Types: BinRel1 = BinRel2 =binary_relation()Returns thestrictrelationcorresponding 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)->Set2Types: 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 theimageof 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). Inimages/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)->Set3Types: Set1 = Set2 = Set3 =a_set()Returns thesymmetricdifference(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)->ExternalSetTypes: ExternalSet =external_set()AnySet =anyset()Returns theexternalsetof an atomic, ordered or unordered set.to_sets(ASet)->SetsTypes: 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)->TypeTypes: AnySet =anyset()Type =type()Returns thetypeof an atomic, ordered or unordered set.union(SetOfSets)->SetTypes: Set =a_set()SetOfSets =set_of_sets()Returns theunionof the set of sets SetOfSets.union(Set1,Set2)->Set3Types: Set1 = Set2 = Set3 =a_set()Returns theunionof Set1 and Set2.union_of_family(Family)->SetTypes: Family =family()Set =a_set()Returns the union of thefamilyFamily. 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)->BinRel2Types: BinRel1 = BinRel2 =binary_relation()Returns a subset S of theweakrelationW corresponding to the binary relation BinRel1. Let F be thefieldof 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)