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**NAME**

sofs - Functions for manipulating sets of sets.

**DESCRIPTION**

This module provides 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 is always the case in this module), this is denoted 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. * Theemptysetcontains no elements. Set A isequalto set B if they contain the same elements, which is denoted A = B. Two ordered sets are equal if they contain the same number of elements and have equal elements at each coordinate. Set B is asubsetof set A if A contains all elements that B contains. 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. As relations are sets, the definitions of the last item (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, theimageof A under R is the set {y : x R y for some x in A}. If B is a subset of Y, 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, 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 R is called 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. Conversely, 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 is not required that the domain of F is 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. As functions are relations, the definitions of the last item (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, we call x afamilyofsubsetsof X. If x is a family of subsets of X, 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 are considered are families of subsets of some set X; in the following, the word "family" is 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, 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, thecanonicalmapis the function that maps every element of X onto its equivalence class. * Relations as defined above (as sets of ordered pairs) are from now on 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 are represented by elements of the relation Sets, which is 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 sets represented by Sets are the elements of the range of 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 are identified with the sets they represent. For example, 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 is that Set(U) is the union of Set(S1) and Set(S2). The types are used to implement the various conditions that sets must 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 used 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, as 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 can change in future versions of this module. 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/2,is_empty_set/1,is_set/1,is_sofs_set/1,to_external/1type/1. 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, 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 specified,[{atom,atom}]is used as the function type.canonical_relation(SetOfSets)->BinRelTypes: BinRel =binary_relation()SetOfSets =set_of_sets()Returns the binary relation containing the elements (E, Set) such that Set belongs toSetOfSetsand E belongs to Set. IfSetOfSetsis apartitionof a set X and R is the equivalence relation in X induced bySetOfSets, 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 functionsFunction1andFunction2. 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 setSetontoAnySet. 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 relationBinRel1. 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 setsSet1andSet2.digraph_to_family(Graph)->Familydigraph_to_family(Graph,Type)->FamilyTypes: Graph =digraph:graph()Family =family()Type =type()Creates afamilyfrom the directed graphGraph. Each vertex a ofGraphis represented by a pair (a, {b[1], ..., b[n]}), where the b[i]:s are the out- neighbors of a. If no type is explicitly specified, [{atom, [atom]}] is used as type of the family. It is assumed thatTypeis 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 relationBinRel. 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 relationBinRel1and therestrictionofBinRel1toSet. 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 ofSet1containing those elements that do not give an element inSet2as the result of applyingSetFun. 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 theextensionofBinRel1such that for each element E inSetthat does not belong to thedomainofBinRel1,BinRel2contains 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 specified,[{atom,[atom]}]is used as the family type.family_difference(Family1,Family2)->Family3Types: Family1 = Family2 = Family3 =family()IfFamily1andFamily2arefamilies, thenFamily3is the family such that the index set is equal to the index set ofFamily1, andFamily3[i] is the difference betweenFamily1[i] andFamily2[i] ifFamily2maps i, otherwiseFamily1[i]. 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()IfFamily1is afamilyandFamily1[i] is a binary relation for every i in the index set ofFamily1, thenFamily2is the family with the same index set asFamily1such thatFamily2[i] is thedomainofFamily1[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()IfFamily1is afamilyandFamily1[i] is a binary relation for every i in the index set ofFamily1, thenFamily2is the family with the same index set asFamily1such thatFamily2[i] is thefieldofFamily1[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()IfFamily1is afamilyandFamily1[i] is a set of sets for every i in the index set ofFamily1, thenFamily2is the family with the same index set asFamily1such thatFamily2[i] is theintersectionofFamily1[i]. IfFamily1[i] is an empty set for some i, 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()IfFamily1andFamily2arefamilies, thenFamily3is the family such that the index set is the intersection ofFamily1:s andFamily2:s index sets, andFamily3[i] is the intersection ofFamily1[i] andFamily2[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()IfFamily1is afamily, thenFamily2is the family with the same index set asFamily1such thatFamily2[i] is the result of callingSetFunwithFamily1[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()IfFamily1is afamilyandFamily1[i] is a binary relation for every i in the index set ofFamily1, thenFamily2is the family with the same index set asFamily1such thatFamily2[i] is therangeofFamily1[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()IfFamily1is afamily, thenFamily2is therestrictionofFamily1to those elements i of the index set for whichFunapplied toFamily1[i] returnstrue. IfFunis a tuple{external,Fun2}, thenFun2is applied to theexternalsetofFamily1[i], otherwiseFunis applied toFamily1[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:graph()Family =family()GraphType = [digraph:d_type()] Creates a directed graph fromfamilyFamily. For each pair (a, {b[1], ..., b[n]}) ofFamily, vertex a and the edges (a, b[i]) for 1 <= i <= n are added to a newly created directed graph. If no graph type is specified,digraph:new/0is used for creating the directed graph, otherwise argumentGraphTypeis 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()IfFamilyis afamily, thenBinRelis the binary relation containing all pairs (i, x) such that i belongs to the index set ofFamilyand x belongs toFamily[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()IfFamily1is afamilyandFamily1[i] is a set of sets for each i in the index set ofFamily1, thenFamily2is the family with the same index set asFamily1such thatFamily2[i] is theunionofFamily1[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()IfFamily1andFamily2arefamilies, thenFamily3is the family such that the index set is the union ofFamily1:s andFamily2:s index sets, andFamily3[i] is the union ofFamily1[i] andFamily2[i] if both map i, otherwiseFamily1[i] orFamily2[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_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 relationBinRel. 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 theexternalsetExternalSetand thetypeType. It is assumed thatTypeis avalidtypeofExternalSet.from_sets(ListOfSets)->SetTypes: Set =a_set()ListOfSets = [anyset()] Returns theunorderedsetcontaining the sets of listListOfSets. 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 tupleTupleOfSets.from_term(Term)->AnySetfrom_term(Term,Type)->AnySetTypes: AnySet =anyset()Term = term() Type =type()Creates an element ofSetsby traversing termTerm, sorting lists, removing duplicates, and deriving or verifying avalidtypefor the so obtained external set. An explicitly specifiedtypeTypecan be used to limit the depth of the traversal; an atomic type stops the traversal, as shown by the following 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, as all functions in this module thatdoanything operate on unordered sets. Creating unordered sets from a collection of ordered sets can 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. The following example shows 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 setSet1under the binary relationBinRel. 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 setsSetOfSets. Intersecting an empty set of sets exits the process with abadargmessage.intersection(Set1,Set2)->Set3Types: Set1 = Set2 = Set3 =a_set()Returns theintersectionofSet1andSet2.intersection_of_family(Family)->SetTypes: Family =family()Set =a_set()Returns the intersection offamilyFamily. 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 functionFunction1. 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 theinverseimageofSet1under the binary relationBinRel. 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 relationBinRelis afunctionor the untyped empty set, otherwisefalse.is_disjoint(Set1,Set2)->BoolTypes: Bool = boolean() Set1 = Set2 =a_set()ReturnstrueifSet1andSet2aredisjoint, otherwisefalse.is_empty_set(AnySet)->BoolTypes: AnySet =anyset()Bool = boolean() ReturnstrueifAnySetis an empty unordered set, otherwisefalse.is_equal(AnySet1,AnySet2)->BoolTypes: AnySet1 = AnySet2 =anyset()Bool = boolean() ReturnstrueifAnySet1andAnySet2areequal, otherwisefalse. The following 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() ReturnstrueifAnySetis anunorderedset, andfalseifAnySetis an ordered set or an atomic set.is_sofs_set(Term)->BoolTypes: Bool = boolean() Term = term() ReturnstrueifTermis anunorderedset, an ordered set, or an atomic set, otherwisefalse.is_subset(Set1,Set2)->BoolTypes: Bool = boolean() Set1 = Set2 =a_set()ReturnstrueifSet1is asubsetofSet2, otherwisefalse.is_type(Term)->BoolTypes: Bool = boolean() Term = term() Returnstrueif termTermis atype.join(Relation1,I,Relation2,J)->Relation3Types: Relation1 = Relation2 = Relation3 =relation()I = J = integer() >= 1 Returns thenaturaljoinof the relationsRelation1andRelation2on coordinatesIandJ. 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()IfTupleOfBinRelsis a non-empty tuple {R[1], ..., R[n]} of binary relations andBinRel1is a binary relation, thenBinRel2is themultiplerelativeproductof the ordered set (R[i], ..., R[n]) andBinRel1. 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 setASet.partition(SetOfSets)->PartitionTypes: SetOfSets =set_of_sets()Partition =a_set()Returns thepartitionof the union of the set of setsSetOfSetssuch that two elements are considered equal if they belong to the same elements ofSetOfSets. 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 thepartitionofSetsuch that two elements are considered equal if the results of applyingSetFunare 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 apartitionofSet1. If the result of applyingSetFunto an element ofSet1gives an element inSet2, the element belongs toSet3, otherwise the element belongs toSet4. 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()ReturnsfamilyFamilywhere the indexed set is apartitionofSetsuch that two elements are considered equal if the results of applyingSetFunare the same value i. This i is the index thatFamilymaps 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 setsTupleOfSets. If (x[1], ..., x[n]) is an element of the n-ary relationRelation, then x[i] is drawn from element i ofTupleOfSets. 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 theCartesianproductofSet1andSet2. 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 ofSet1by the result of applyingSetFunto the element. IfSetFunis a number i >= 1 andSet1is a relation, then the returned set is theprojectionofSet1onto 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 relationBinRel. 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. IfTypeis an integer N, then[{atom,...,atom}]), where the tuple size is N, is used as type of the relation. If no type is explicitly specified, the size of the first tuple ofTuplesis used if there is such a tuple.relation([])is equivalent torelation([],2).relation_to_family(BinRel)->FamilyTypes: Family =family()BinRel =binary_relation()ReturnsfamilyFamilysuch that the index set is equal to thedomainof the binary relationBinRel, andFamily[i] is theimageof the set of i underBinRel. 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()IfListOfBinRelsis a non-empty list [R[1], ..., R[n]] of binary relations andBinRel1is a binary relation, thenBinRel2is therelativeproductof the ordered set (R[i], ..., R[n]) andBinRel1. IfBinRel1is 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}}] Notice 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 relationsBinRel1andBinRel2.relative_product1(BinRel1,BinRel2)->BinRel3Types: BinRel1 = BinRel2 = BinRel3 =binary_relation()Returns therelativeproductof theconverseof the binary relationBinRel1and the binary relationBinRel2. 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 relationBinRel1toSet. 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 ofSet1containing those elements that gives an element inSet2as the result of applyingSetFun. 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 specified,[atom]is used as the set type.specification(Fun,Set1)->Set2Types: Fun =spec_fun()Set1 = Set2 =a_set()Returns the set containing every element ofSet1for whichFunreturnstrue. IfFunis a tuple{external,Fun2},Fun2is applied to theexternalsetof each element, otherwiseFunis 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 relationBinRel1. 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 isSet1. The value of an element of the domain is the result of applyingSetFunto 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}] LetSetOfSetsbe a set of sets andBinRela binary relation. The function that maps each elementSetofSetOfSetsonto theimageofSetunderBinRelis returned by the following function: images(SetOfSets, BinRel) -> Fun = fun(Set) -> sofs:image(BinRel, Set) end, sofs:substitution(Fun, SetOfSets). External unordered sets are represented as sorted lists. So, creating the image of a set under a relation R can traverse all elements of R (to that comes the sorting of results, the image). Inimage/2,BinRelis traversed once for each element ofSetOfSets, which can take too long. The following efficient function can be used instead under the assumption that the image of each element ofSetOfSetsunderBinRelis 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) ofSet1andSet2. 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: *Set3contains the elements ofSet1that do not belong toSet2. *Set4contains the elements ofSet1that belong toSet2. *Set5contains the elements ofSet2that do not belong toSet1.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 setASetas a tuple of sets, and the elements of the unordered setASetas 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 setsSetOfSets.union(Set1,Set2)->Set3Types: Set1 = Set2 = Set3 =a_set()Returns theunionofSet1andSet2.union_of_family(Family)->SetTypes: Family =family()Set =a_set()Returns the union offamilyFamily. 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 relationBinRel1. Let F be thefieldofBinRel1. 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)