sofs.3erl - Man Page

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)}.

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:

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/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.

Data Types

anyset() = ordset() | a_set()

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

binary_relation() = relation()

A binary relation.

external_set() = term()

An external set.

family() = a_function()

A family (of subsets).

a_function() = relation()

A function.

ordset()

An ordered set.

relation() = a_set()

An n-ary relation.

a_set()

An unordered set.

set_of_sets() = a_set()

An unordered set of unordered sets.

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

A SetFun.

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

A type.

tuple_of(T)

A tuple where the elements are of type T.

Exports

a_function(Tuples) -> Function
a_function(Tuples, Type) -> Function
Types:

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

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

canonical_relation(SetOfSets) -> BinRel
Types:

BinRel = binary_relation()
SetOfSets = set_of_sets()

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

1> Ss = sofs:from_term([[a,b],[b,c]]),
CR = sofs:canonical_relation(Ss),
sofs:to_external(CR).
[{a,[a,b]},{b,[a,b]},{b,[b,c]},{c,[b,c]}]
composite(Function1, Function2) -> Function3
Types:

Function1 = Function2 = Function3 = a_function()

Returns the composite of the functions Function1 and Function2.

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

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

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

1> S = sofs:set([a,b]),
E = sofs:from_term(1),
R = sofs:constant_function(S, E),
sofs:to_external(R).
[{a,1},{b,1}]
converse(BinRel1) -> BinRel2
Types:

BinRel1 = BinRel2 = binary_relation()

Returns the converse of the binary relation BinRel1.

1> R1 = sofs:relation([{1,a},{2,b},{3,a}]),
R2 = sofs:converse(R1),
sofs:to_external(R2).
[{a,1},{a,3},{b,2}]
difference(Set1, Set2) -> Set3
Types:

Set1 = Set2 = Set3 = a_set()

Returns the difference of the sets Set1 and Set2.

digraph_to_family(Graph) -> Family
digraph_to_family(Graph, Type) -> Family
Types:

Graph = digraph: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. If no type is explicitly specified, [{atom, [atom]}] is used as type of the family. It is assumed that Type is a valid type of the external set of the family.

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

domain(BinRel) -> Set
Types:

BinRel = binary_relation()
Set = a_set()

Returns the domain of the binary relation BinRel.

1> R = sofs:relation([{1,a},{1,b},{2,b},{2,c}]),
S = sofs:domain(R),
sofs:to_external(S).
[1,2]
drestriction(BinRel1, Set) -> BinRel2
Types:

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

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

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

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

drestriction(SetFun, Set1, Set2) -> Set3
Types:

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

Returns a subset of Set1 containing those elements that do not 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)).

empty_set() -> Set
Types:

Set = a_set()

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

extension(BinRel1, Set, AnySet) -> BinRel2
Types:

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

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

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

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

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

family_difference(Family1, Family2) -> Family3
Types:

Family1 = Family2 = Family3 = family()

If Family1 and Family2 are families, then Family3 is the family such that the index set is equal to the index set of Family1, and Family3[i] is the difference between Family1[i] and Family2[i] if Family2 maps i, 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]}]
family_domain(Family1) -> Family2
Types:

Family1 = Family2 = family()

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

1> FR = sofs:from_term([{a,[{1,a},{2,b},{3,c}]},{b,[]},{c,[{4,d},{5,e}]}]),
F = sofs:family_domain(FR),
sofs:to_external(F).
[{a,[1,2,3]},{b,[]},{c,[4,5]}]
family_field(Family1) -> Family2
Types:

Family1 = Family2 = family()

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

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

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

family_intersection(Family1) -> Family2
Types:

Family1 = Family2 = family()

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

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

1> F1 = sofs:from_term([{a,[[1,2,3],[2,3,4]]},{b,[[x,y,z],[x,y]]}]),
F2 = sofs:family_intersection(F1),
sofs:to_external(F2).
[{a,[2,3]},{b,[x,y]}]
family_intersection(Family1, Family2) -> Family3
Types:

Family1 = Family2 = Family3 = family()

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

1> F1 = sofs:family([{a,[1,2]},{b,[3,4]},{c,[5,6]}]),
F2 = sofs:family([{b,[4,5]},{c,[7,8]},{d,[9,10]}]),
F3 = sofs:family_intersection(F1, F2),
sofs:to_external(F3).
[{b,[4]},{c,[]}]
family_projection(SetFun, Family1) -> Family2
Types:

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

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

1> F1 = sofs:from_term([{a,[[1,2],[2,3]]},{b,[[]]}]),
F2 = sofs:family_projection(fun sofs:union/1, F1),
sofs:to_external(F2).
[{a,[1,2,3]},{b,[]}]
family_range(Family1) -> Family2
Types:

Family1 = Family2 = family()

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

1> FR = sofs:from_term([{a,[{1,a},{2,b},{3,c}]},{b,[]},{c,[{4,d},{5,e}]}]),
F = sofs:family_range(FR),
sofs:to_external(F).
[{a,[a,b,c]},{b,[]},{c,[d,e]}]
family_specification(Fun, Family1) -> Family2
Types:

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

If Family1 is a family, then Family2 is the restriction of Family1 to those elements i of the index set for which Fun applied to Family1[i] returns true. If Fun is a tuple {external, Fun2}, 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]}]
family_to_digraph(Family) -> Graph
family_to_digraph(Family, GraphType) -> Graph
Types:

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.

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

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

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

family_to_relation(Family) -> BinRel
Types:

Family = family()
BinRel = binary_relation()

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

1> F = sofs:family([{a,[]}, {b,[1]}, {c,[2,3]}]),
R = sofs:family_to_relation(F),
sofs:to_external(R).
[{b,1},{c,2},{c,3}]
family_union(Family1) -> Family2
Types:

Family1 = Family2 = family()

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

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

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

family_union(Family1, Family2) -> Family3
Types:

Family1 = Family2 = Family3 = family()

If Family1 and Family2 are families, then Family3 is the family such that the index set is the union of Family1:s and Family2:s index sets, and Family3[i] is the union of Family1[i] and Family2[i] if both 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]}]
field(BinRel) -> Set
Types:

BinRel = binary_relation()
Set = a_set()

Returns the field of the binary relation BinRel.

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

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

from_external(ExternalSet, Type) -> AnySet
Types:

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

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

from_sets(ListOfSets) -> Set
Types:

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

Returns the unordered set containing the sets of list ListOfSets.

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

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

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

from_term(Term) -> AnySet
from_term(Term, Type) -> AnySet
Types:

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

Creates an element of Sets by traversing 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.

image(BinRel, Set1) -> Set2
Types:

BinRel = binary_relation()
Set1 = 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]
intersection(SetOfSets) -> Set
Types:

Set = a_set()
SetOfSets = set_of_sets()

Returns the intersection of the set of sets SetOfSets.

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

intersection(Set1, Set2) -> Set3
Types:

Set1 = Set2 = Set3 = a_set()

Returns the intersection of Set1 and Set2.

intersection_of_family(Family) -> Set
Types:

Family = family()
Set = a_set()

Returns the intersection of family Family.

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

1> F = sofs:family([{a,[0,2,4]},{b,[0,1,2]},{c,[2,3]}]),
S = sofs:intersection_of_family(F),
sofs:to_external(S).
[2]
inverse(Function1) -> Function2
Types:

Function1 = Function2 = a_function()

Returns the inverse of function Function1.

1> R1 = sofs:relation([{1,a},{2,b},{3,c}]),
R2 = sofs:inverse(R1),
sofs:to_external(R2).
[{a,1},{b,2},{c,3}]
inverse_image(BinRel, Set1) -> Set2
Types:

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

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

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

Bool = boolean()
BinRel = binary_relation()

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

is_disjoint(Set1, Set2) -> Bool
Types:

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

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

is_empty_set(AnySet) -> Bool
Types:

AnySet = anyset()
Bool = boolean()

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

is_equal(AnySet1, AnySet2) -> Bool
Types:

AnySet1 = 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).
true
is_set(AnySet) -> Bool
Types:

AnySet = anyset()
Bool = boolean()

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

is_sofs_set(Term) -> Bool
Types:

Bool = boolean()
Term = term()

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

is_subset(Set1, Set2) -> Bool
Types:

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

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

is_type(Term) -> Bool
Types:

Bool = boolean()
Term = term()

Returns true if term Term is a type.

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

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

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

1> R1 = sofs:relation([{a,x,1},{b,y,2}]),
R2 = sofs:relation([{1,f,g},{1,h,i},{2,3,4}]),
J = sofs:join(R1, 3, R2, 1),
sofs:to_external(J).
[{a,x,1,f,g},{a,x,1,h,i},{b,y,2,3,4}]
multiple_relative_product(TupleOfBinRels, BinRel1) -> BinRel2
Types:

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

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

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

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

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

partition(SetOfSets) -> Partition
Types:

SetOfSets = set_of_sets()
Partition = a_set()

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

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

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

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

1> Ss = sofs:from_term([[a],[b],[c,d],[e,f]]),
SetFun = fun(S) -> sofs:from_term(sofs:no_elements(S)) end,
P = sofs:partition(SetFun, Ss),
sofs:to_external(P).
[[[a],[b]],[[c,d],[e,f]]]
partition(SetFun, Set1, Set2) -> {Set3, Set4}
Types:

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

Returns a pair of sets that, regarded as constituting a set, forms a partition of Set1. If the result of applying SetFun to an element of Set1 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)}.

partition_family(SetFun, Set) -> Family
Types:

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}]}]
product(TupleOfSets) -> Relation
Types:

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

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

1> S1 = sofs:set([a,b]),
S2 = sofs:set([1,2]),
S3 = sofs:set([x,y]),
P3 = sofs:product({S1,S2,S3}),
sofs:to_external(P3).
[{a,1,x},{a,1,y},{a,2,x},{a,2,y},{b,1,x},{b,1,y},{b,2,x},{b,2,y}]
product(Set1, Set2) -> BinRel
Types:

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

Returns the Cartesian product of Set1 and Set2.

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

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

projection(SetFun, Set1) -> Set2
Types:

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

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

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

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

BinRel = binary_relation()
Set = a_set()

Returns the range of the binary relation BinRel.

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

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

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

relation_to_family(BinRel) -> Family
Types:

Family = family()
BinRel = binary_relation()

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

1> R = sofs:relation([{b,1},{c,2},{c,3}]),
F = sofs:relation_to_family(R),
sofs:to_external(F).
[{b,[1]},{c,[2,3]}]
relative_product(ListOfBinRels) -> BinRel2
relative_product(ListOfBinRels, BinRel1) -> BinRel2
Types:

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

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

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

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

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.

relative_product(BinRel1, BinRel2) -> BinRel3
Types:

BinRel1 = BinRel2 = BinRel3 = binary_relation()

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

relative_product1(BinRel1, BinRel2) -> BinRel3
Types:

BinRel1 = BinRel2 = BinRel3 = binary_relation()

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

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

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

restriction(BinRel1, Set) -> BinRel2
Types:

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

Returns the restriction of the binary relation BinRel1 to Set.

1> R1 = sofs:relation([{1,a},{2,b},{3,c}]),
S = sofs:set([1,2,4]),
R2 = sofs:restriction(R1, S),
sofs:to_external(R2).
[{1,a},{2,b}]
restriction(SetFun, Set1, Set2) -> Set3
Types:

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

Returns a subset of Set1 containing those elements that 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}]
set(Terms) -> Set
set(Terms, Type) -> Set
Types:

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

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

specification(Fun, Set1) -> Set2
Types:

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

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

1> R1 = sofs:relation([{a,1},{b,2}]),
R2 = sofs:relation([{x,1},{x,2},{y,3}]),
S1 = sofs:from_sets([R1,R2]),
S2 = sofs:specification(fun sofs:is_a_function/1, S1),
sofs:to_external(S2).
[[{a,1},{b,2}]]
strict_relation(BinRel1) -> BinRel2
Types:

BinRel1 = BinRel2 = binary_relation()

Returns the strict relation corresponding to the binary relation BinRel1.

1> R1 = sofs:relation([{1,1},{1,2},{2,1},{2,2}]),
R2 = sofs:strict_relation(R1),
sofs:to_external(R2).
[{1,2},{2,1}]
substitution(SetFun, Set1) -> Set2
Types:

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

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

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

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

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

Let SetOfSets be a set of sets and BinRel a binary relation. The function that maps each element Set of SetOfSets onto the image of Set under BinRel is returned by 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).
symdiff(Set1, Set2) -> Set3
Types:

Set1 = Set2 = Set3 = a_set()

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

1> S1 = sofs:set([1,2,3]),
S2 = sofs:set([2,3,4]),
P = sofs:symdiff(S1, S2),
sofs:to_external(P).
[1,4]
symmetric_partition(Set1, Set2) -> {Set3, Set4, Set5}
Types:

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

Returns a triple of sets:

  • Set3 contains the elements of Set1 that do not belong to Set2.
  • Set4 contains the elements of Set1 that belong to Set2.
  • Set5 contains the elements of Set2 that do not belong to Set1.
to_external(AnySet) -> ExternalSet
Types:

ExternalSet = external_set()
AnySet = anyset()

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

to_sets(ASet) -> Sets
Types:

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

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

type(AnySet) -> Type
Types:

AnySet = anyset()
Type = type()

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

union(SetOfSets) -> Set
Types:

Set = a_set()
SetOfSets = set_of_sets()

Returns the union of the set of sets SetOfSets.

union(Set1, Set2) -> Set3
Types:

Set1 = Set2 = Set3 = a_set()

Returns the union of Set1 and Set2.

union_of_family(Family) -> Set
Types:

Family = family()
Set = a_set()

Returns the union of family Family.

1> F = sofs:family([{a,[0,2,4]},{b,[0,1,2]},{c,[2,3]}]),
S = sofs:union_of_family(F),
sofs:to_external(S).
[0,1,2,3,4]
weak_relation(BinRel1) -> BinRel2
Types:

BinRel1 = BinRel2 = binary_relation()

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

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

See Also

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

Info

stdlib 3.14 Ericsson AB Erlang Module Definition