View Source sofs (stdlib v6.0)
Functions for manipulating sets of sets.
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.
The data representing sofs as used by this module is to be regarded as opaque
by other modules. In abstract terms, the representation is a composite type of
existing Erlang terms. See note on
data types. Any code assuming
knowledge of the format is running on thin ice.
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)}.
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 empty set contains no elements.
Set A is equal to 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 a subset of set A if A contains all elements that B contains.
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 × 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 × 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.
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, the image of A under R is the set {y : x R y for some x in A}. If B is a subset of Y, 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, 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 R is called 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. 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 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 × 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 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, we call x a family of subsets of X.
If x is a family of subsets of X, 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 are considered are families of subsets of some set X; in the following, the word "family" is 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, 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, 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) are from now on 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[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] × ... × 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[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 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 (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/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 calling
union/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}
2The 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/1 type/1.
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, operator ==/2 is used.
See Also
Summary
Types
An unordered set.
Any kind of set (also included are the atomic sets).
An ordered set.
An unordered set of unordered sets.
A tuple where the elements are of type T.
Functions
Equivalent to a_function(Tuples, [{atom, atom}]).
Creates a function.
Returns the binary relation containing the elements (E, Set) such that Set
belongs to SetOfSets and E belongs to Set.
Returns the composite of the functions Function1 and
Function2.
Creates the function that maps each element of set Set
onto AnySet.
Returns the converse of the binary relation BinRel1.
Returns the difference of the sets Set1 and Set2.
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-neighbors of a. It is assumed that Type is
a valid type of the external set of the family.
Returns the domain of the binary relation BinRel.
Returns the difference between the binary relation BinRel1 and the
restriction of BinRel1 to Set.
Returns a subset of Set1 containing those elements that do not give an element
in Set2 as the result of applying SetFun.
Returns the untyped empty set. empty_set/0 is
equivalent to from_term([], ['_']).
Equivalent to family(Tuples, [{atom, [atom]}]).
Creates a family of subsets. family(F, T) is
equivalent to from_term(F, T) if the result is a 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, otherwise Family1[i].
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 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].
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.
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}, then Fun2 is applied to the
external set of Family1[i], otherwise Fun is
applied to Family1[i].
Equivalent to family_to_digraph(Family, []).
Creates a directed graph from family Family. For each pair
(a, {b[1], ..., b[n]}) of Family, vertex a and the edges (a, b[i]) for
1 <= i <= n are added to a newly created directed graph.
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].
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 map i, otherwise Family1[i] or Family2[i].
Returns the field of the binary relation BinRel.
Creates a set from the external set ExternalSet and
the type Type. It is assumed that Type is a
valid type of ExternalSet.
Returns the unordered set containing the sets of
list ListOfSets.
Equivalent to from_term(Term, '_').
Creates an element of Sets by
traversing term Term, sorting lists, removing duplicates, and deriving or
verifying a valid type for the so obtained external set.
Returns the image of set Set1 under the binary relation
BinRel.
Returns the intersection of the set of sets
SetOfSets.
Returns the intersection of Set1 and Set2.
Returns the intersection of family Family.
Returns the inverse of function Function1.
Returns the inverse image of Set1 under the binary
relation BinRel.
Returns true if the binary relation BinRel is a
function or the untyped empty set, otherwise false.
Returns true if Set1 and Set2 are disjoint, otherwise
false.
Returns true if AnySet is an empty unordered set, otherwise false.
Returns true if AnySet1 and AnySet2 are equal, otherwise
false. The following example shows that ==/2 is used when comparing sets for
equality
Returns true if AnySet appears to be an
unordered set, and false if AnySet is an ordered
set or an atomic set or any other term.
Returns true if Term appears to be an
unordered set, an ordered set, or an atomic set,
otherwise false.
Returns true if Set1 is a subset of Set2, otherwise
false.
Returns true if term Term is a type.
Returns the natural join of the relations Relation1
and Relation2 on coordinates I and J.
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.
Returns the number of elements of the ordered or unordered set ASet.
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.
Returns the partition of Set such that two elements are
considered equal if the results of applying SetFun are equal.
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 gives an element in Set2, the element belongs to Set3,
otherwise the element belongs to Set4.
Returns 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.
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.
Returns the Cartesian product of Set1 and
Set2.
Returns the set created by substituting each element of Set1 by the result of
applying SetFun to the element.
Returns the range of the binary relation BinRel.
Equivalent to relation(Tuples, Type) where Type is the size
of the first tuple of Tuples is used if there is such a 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.
Returns the relative product of the
converse of the binary relation BinRel1 and the binary
relation BinRel2.
Equivalent to relative_product/2.
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.
Returns the restriction of the binary relation BinRel1
to Set.
Returns a subset of Set1 containing those elements that gives an element in
Set2 as the result of applying SetFun.
Equivalent to set(Terms, [atom]).
Creates an unordered set. set(L, T) is
equivalent to from_term(L, T), if the result is an unordered
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.
Returns the strict relation corresponding to the
binary relation BinRel1.
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.
Returns the symmetric difference (or the
Boolean sum) of Set1 and Set2.
Returns a triple of sets
Returns the external set of an atomic, ordered, or unordered set.
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.
Returns the type of an atomic, ordered, or unordered set.
Returns the union of the set of sets SetOfSets.
Returns the union of Set1 and Set2.
Returns the union of family Family.
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.
Types
-type a_function() :: relation().
A function.
-opaque a_set()
An unordered set.
Any kind of set (also included are the atomic sets).
-type binary_relation() :: relation().
-type external_set() :: term().
An external set.
-type family() :: a_function().
A family (of subsets).
-opaque ordset()
An ordered set.
-type relation() :: a_set().
An n-ary relation.
-type set_fun() :: pos_integer() | {external, fun((external_set()) -> external_set())} | fun((anyset()) -> anyset()).
A SetFun.
-type set_of_sets() :: a_set().
An unordered set of unordered sets.
-type spec_fun() :: {external, fun((external_set()) -> boolean())} | fun((anyset()) -> boolean()).
-type tuple_of(_T) :: tuple().
A tuple where the elements are of type T.
-type type() :: term().
A type.
Functions
-spec a_function(Tuples) -> Function when Function :: a_function(), Tuples :: [tuple()].
Equivalent to a_function(Tuples, [{atom, atom}]).
-spec a_function(Tuples, Type) -> Function when 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.
-spec canonical_relation(SetOfSets) -> BinRel when 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]}]-spec composite(Function1, Function2) -> Function3 when Function1 :: a_function(), Function2 :: a_function(), 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}]-spec constant_function(Set, AnySet) -> Function when AnySet :: anyset(), Function :: a_function(), Set :: a_set().
Creates the function that maps each element of 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}]-spec converse(BinRel1) -> BinRel2 when BinRel1 :: binary_relation(), 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}]Returns the difference of the sets Set1 and Set2.
-spec digraph_to_family(Graph) -> Family when Graph :: digraph:graph(), Family :: family().
Equivalent to digraph_to_family(Graph, [{atom, [atom]}]).
-spec digraph_to_family(Graph, Type) -> Family when Graph :: digraph:graph(), 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-neighbors of a. 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)).
-spec domain(BinRel) -> Set when 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]-spec drestriction(BinRel1, Set) -> BinRel2 when BinRel1 :: binary_relation(), 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)).
-spec drestriction(SetFun, Set1, Set2) -> Set3 when SetFun :: set_fun(), Set1 :: a_set(), Set2 :: a_set(), Set3 :: a_set().
Returns a subset of Set1 containing those elements that do not give 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)).
-spec empty_set() -> Set when Set :: a_set().
Returns the untyped empty set. empty_set/0 is
equivalent to from_term([], ['_']).
-spec extension(BinRel1, Set, AnySet) -> BinRel2 when AnySet :: anyset(), BinRel1 :: binary_relation(), 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,[]}]Equivalent to family(Tuples, [{atom, [atom]}]).
Creates a family of subsets. family(F, T) is
equivalent to from_term(F, T) if the result is a family.
-spec family_difference(Family1, Family2) -> Family3 when Family1 :: family(), Family2 :: family(), 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, otherwise Family1[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]}]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]}]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)).
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, 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]}]-spec family_intersection(Family1, Family2) -> Family3 when Family1 :: family(), Family2 :: family(), 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,[]}]-spec family_projection(SetFun, Family1) -> Family2 when SetFun :: set_fun(), Family1 :: family(), 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,[]}]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]}]-spec family_specification(Fun, Family1) -> Family2 when Fun :: spec_fun(), Family1 :: family(), 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}, then 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]}]-spec family_to_digraph(Family) -> Graph when Graph :: digraph:graph(), Family :: family().
Equivalent to family_to_digraph(Family, []).
-spec family_to_digraph(Family, GraphType) -> Graph when Graph :: digraph:graph(), Family :: family(), GraphType :: [digraph:d_type()].
Creates a directed graph from family Family. For each pair
(a, {b[1], ..., b[n]}) of Family, vertex a and the edges (a, b[i]) for
1 <= i <= n are added to a newly created directed graph.
GraphType is passed on 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.
-spec family_to_relation(Family) -> BinRel when 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}]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).
-spec family_union(Family1, Family2) -> Family3 when Family1 :: family(), Family2 :: family(), 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 map i, otherwise Family1[i] or 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_union(F1, F2),
sofs:to_external(F3).
[{a,[1,2]},{b,[3,4,5]},{c,[5,6,7,8]},{d,[9,10]}]-spec field(BinRel) -> Set when 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)).
-spec from_external(ExternalSet, Type) -> AnySet when 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.
-spec from_sets(ListOfSets) -> Set when Set :: a_set(), ListOfSets :: [anyset()]; (TupleOfSets) -> Ordset when Ordset :: ordset(), TupleOfSets :: tuple_of(anyset()).
Returns the unordered set containing the sets of
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}]]Returns the ordered set containing the sets of the
non-empty tuple TupleOfSets.
Equivalent to from_term(Term, '_').
Creates an element of Sets by
traversing term Term, sorting lists, removing duplicates, and deriving or
verifying a valid type for the so obtained external set.
An explicitly specified type Type can 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_term can 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 that do anything 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 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.
-spec image(BinRel, Set1) -> Set2 when BinRel :: binary_relation(), Set1 :: a_set(), Set2 :: a_set().
Returns the image of 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]-spec intersection(SetOfSets) -> Set when 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.
Returns the intersection of Set1 and Set2.
Returns the intersection of 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]-spec inverse(Function1) -> Function2 when Function1 :: a_function(), Function2 :: a_function().
Returns the inverse of 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}]-spec inverse_image(BinRel, Set1) -> Set2 when BinRel :: binary_relation(), Set1 :: a_set(), 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]-spec is_a_function(BinRel) -> Bool when Bool :: boolean(), BinRel :: binary_relation().
Returns true if the binary relation BinRel is a
function or the untyped empty set, otherwise false.
Returns true if Set1 and Set2 are disjoint, otherwise
false.
Returns true if AnySet is an empty unordered set, otherwise false.
-spec is_equal(AnySet1, AnySet2) -> Bool when AnySet1 :: anyset(), AnySet2 :: anyset(), Bool :: boolean().
Returns true if AnySet1 and AnySet2 are equal, otherwise
false. The following 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).
trueReturns true if AnySet appears to be an
unordered set, and false if AnySet is an ordered
set or an atomic set or any other term.
Note that the test is shallow and this function will return true for any term
that coincides with the representation of an unordered set. See also note on
data types.
Returns true if Term appears to be an
unordered set, an ordered set, or an atomic set,
otherwise false.
Note that this function will return true for any term that
coincides with the representation of a sofs set. See also note on
data types.
Returns true if Set1 is a subset of Set2, otherwise
false.
Returns true if term Term is a type.
-spec join(Relation1, I, Relation2, J) -> Relation3 when Relation1 :: relation(), Relation2 :: relation(), Relation3 :: relation(), I :: pos_integer(), J :: pos_integer().
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}]-spec multiple_relative_product(TupleOfBinRels, BinRel1) -> BinRel2 when TupleOfBinRels :: tuple_of(BinRel), BinRel :: binary_relation(), BinRel1 :: binary_relation(), 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}]-spec no_elements(ASet) -> NoElements when ASet :: a_set() | ordset(), NoElements :: non_neg_integer().
Returns the number of elements of the ordered or unordered set ASet.
-spec partition(SetOfSets) -> Partition when 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]]-spec partition(SetFun, Set) -> Partition when SetFun :: set_fun(), Partition :: a_set(), 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]]]-spec partition(SetFun, Set1, Set2) -> {Set3, Set4} when SetFun :: set_fun(), Set1 :: a_set(), Set2 :: a_set(), Set3 :: a_set(), 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 gives 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)}.
-spec partition_family(SetFun, Set) -> Family when Family :: family(), SetFun :: set_fun(), Set :: a_set().
Returns 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}]}]-spec product(TupleOfSets) -> Relation when 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}]-spec product(Set1, Set2) -> BinRel when BinRel :: binary_relation(), Set1 :: a_set(), 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}).
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]-spec range(BinRel) -> Set when 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]Equivalent to relation(Tuples, Type) where Type is the size
of the first tuple of Tuples is used if there is such a tuple.
If tuples is [], then Type is 2.
-spec relation(Tuples, Type) -> Relation when 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 tuple size is N,
is used as type of the relation.
-spec relation_to_family(BinRel) -> Family when Family :: family(), BinRel :: binary_relation().
Returns 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]}]-spec relative_product1(BinRel1, BinRel2) -> BinRel3 when BinRel1 :: binary_relation(), BinRel2 :: binary_relation(), 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).
-spec relative_product(ListOfBinRels) -> BinRel2 when ListOfBinRels :: [BinRel, ...], BinRel :: binary_relation(), BinRel2 :: binary_relation().
Equivalent to relative_product/2.
-spec relative_product(ListOfBinRels, BinRel1) -> BinRel2 when ListOfBinRels :: [BinRel, ...], BinRel :: binary_relation(), BinRel1 :: binary_relation(), BinRel2 :: binary_relation(); (BinRel1, BinRel2) -> BinRel3 when BinRel1 :: binary_relation(), BinRel2 :: binary_relation(), BinRel3 :: 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] × ... × 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 that relative_product([R1], R2) is different
from relative_product(R1, R2); the list of one element
is not identified with the element itself.
Returns the relative product of the binary
relations BinRel1 and BinRel2.
-spec restriction(BinRel1, Set) -> BinRel2 when BinRel1 :: binary_relation(), 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}]-spec restriction(SetFun, Set1, Set2) -> Set3 when SetFun :: set_fun(), Set1 :: a_set(), Set2 :: a_set(), Set3 :: a_set().
Returns a subset of Set1 containing those elements that gives 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}]Equivalent to set(Terms, [atom]).
Creates an unordered set. set(L, T) is
equivalent to from_term(L, T), if the result is an unordered
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}]]-spec strict_relation(BinRel1) -> BinRel2 when BinRel1 :: binary_relation(), 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}]-spec substitution(SetFun, Set1) -> Set2 when SetFun :: set_fun(), Set1 :: a_set(), 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 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). In image/2, BinRel is traversed once for
each element of SetOfSets, which can take too long. The following efficient
function can 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).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]-spec symmetric_partition(Set1, Set2) -> {Set3, Set4, Set5} when Set1 :: a_set(), Set2 :: a_set(), Set3 :: a_set(), Set4 :: a_set(), 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.
-spec to_external(AnySet) -> ExternalSet when ExternalSet :: external_set(), AnySet :: anyset().
Returns the external set of an atomic, ordered, or unordered set.
-spec to_sets(ASet) -> Sets when 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.
Returns the type of an atomic, ordered, or unordered set.
-spec union(SetOfSets) -> Set when Set :: a_set(), SetOfSets :: set_of_sets().
Returns the union of the set of sets SetOfSets.
Returns the union of Set1 and Set2.
Returns the union of 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]-spec weak_relation(BinRel1) -> BinRel2 when BinRel1 :: binary_relation(), 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}]