# sofs

## Functions for Manipulating Sets of Sets

The `sofs`

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

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

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

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

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

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

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

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

Relations as defined above (as sets of ordered pairs) will from
now on be referred to as *binary relations*. We call a
set of ordered sets (x[1], ..., x[n])
an *(n-ary) relation*, and say that the relation is a subset of
the Cartesian product
X[1] × ... × X[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 will be represented by elements of the relation Sets, defined as the smallest set such that:

- for every atom T except '_' and for every term X,
(T, X) belongs to Sets (
*atomic sets*); - (['_'], []) belongs to Sets (the
*untyped empty set*); - for every tuple T = {T[1], ..., T[n]} and
for every tuple X = {X[1], ..., X[n]}, if
(T[i], X[i]) belongs to Sets for every
1 <= i <= n then (T, X) belongs
to Sets (
*ordered sets*); - for every term T, if X is the empty list or a non-empty
sorted list [X[1], ..., X[n]] without duplicates
such that (T, X[i]) belongs to Sets for every
1 <= i <= n, then ([T], X)
belongs to Sets (
*typed unordered sets*).

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

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

- Set(T,Term) = Term, where T is an atom;
- Set({T[1], ..., T[n]}, {X[1], ..., X[n]}) = (Set(T[1], X[1]), ..., Set(T[n], X[n]));
- Set([T], [X[1], ..., X[n]]) = {Set(T, X[1]), ..., Set(T, X[n])};
- Set([T], []) = {}.

When there is no risk of confusion, elements of Sets will be
identified with the sets they represent. For instance, if U is
the result of calling `union/2`

with S1 and S2 as
arguments, then U is said to be the union of S1 and S2. A more
precise formulation would be that Set(U) is the union of Set(S1)
and Set(S2).

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

A few functions of this module (`drestriction/3`

,
`family_projection/2`

, `partition/2`

,
`partition_family/2`

, `projection/2`

,
`restriction/3`

, `substitution/2`

) accept an Erlang
function as a means to modify each element of a given unordered
set. Such a function, called SetFun in the following, can be
specified as a functional object (fun), a tuple
`{external, Fun}`

, or an integer. If SetFun is
specified as a fun, the fun is applied to each element of the
given set and the return value is assumed to be a set. If SetFun
is specified as a tuple `{external, Fun}`

, Fun is applied
to the external set of each element of the given set and the
return value is assumed to be an external set. Selecting the
elements of an unordered set as external sets and assembling a
new unordered set from a list of external sets is in the present
implementation more efficient than modifying each element as a
set. However, this optimization can only be utilized when the
elements of the unordered set are atomic or ordered sets. It
must also be the case that the type of the elements matches some
clause of Fun (the type of the created set is the result of
applying Fun to the type of the given set), and that Fun does
nothing but selecting, duplicating or rearranging parts of the
elements. Specifying a SetFun as an integer I is equivalent to
specifying `{external, fun(X) -> element(I, X) end}`

,
but is to be preferred since it makes it possible to handle this
case even more efficiently. Examples of SetFuns:

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

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

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

,
`is_empty_set`

, `is_set`

, `is_sofs_set`

,
`to_external`

, `type`

.

The functions of this module exit the process with a
`badarg`

, `bad_function`

, or `type_mismatch`

message when given badly formed arguments or sets the types of
which are not compatible.

When comparing external sets the operator `==/2`

is used.

*Types*

anyset() = - an unordered, ordered or atomic set - binary_relation() = - a binary relation - bool() = true | false external_set() = - an external set - family() = - a family (of subsets) - function() = - a function - ordset() = - an ordered set - relation() = - an n-ary relation - set() = - an unordered set - set_of_sets() = - an unordered set of set() - set_fun() = integer() >= 1 | {external, fun(external_set()) -> external_set()} | fun(anyset()) -> anyset() spec_fun() = {external, fun(external_set()) -> bool()} | fun(anyset()) -> bool() type() = - a type -

#### Functions

### a_function(Tuples [, Type]) -> Function

`Function = function()`

`Tuples = [tuple()]`

`Type = type()`

### canonical_relation(SetOfSets) -> BinRel

`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

`Function1 = Function2 = Function3 = 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

`AnySet = anyset()`

`Function = function()`

`Set = set()`

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

1>`S = sofs:set([a,b]),`

`E = sofs:from_term(1),`

`R = sofs:constant_function(S, E),`

`sofs:to_external(R).`

[{a,1},{b,1}]

### converse(BinRel1) -> BinRel2

`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

`Set1 = Set2 = Set3 = set()`

Returns the difference of the sets Set1 and Set2.

### digraph_to_family(Graph [, Type]) -> Family

`Graph = digraph() - see digraph(3) -`

`Family = family()`

`Type = type()`

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

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

.

### domain(BinRel) -> Set

`BinRel = binary_relation()`

`Set = 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

`BinRel1 = BinRel2 = binary_relation()`

`Set = 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

`SetFun = set_fun()`

`Set1 = Set2 = Set3 = set()`

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

1>`SetFun = {external, fun({_A,B,C}) -> {B,C} end},`

`R1 = sofs:relation([{a,aa,1},{b,bb,2},{c,cc,3}]),`

`R2 = sofs:relation([{bb,2},{cc,3},{dd,4}]),`

`R3 = sofs:drestriction(SetFun, R1, R2),`

`sofs:to_external(R3).`

[{a,aa,1}]

`drestriction(F, S1, S2)`

is equivalent to
`difference(S1, restriction(F, S1, S2))`

.

### empty_set() -> Set

`Set = set()`

Returns the untyped empty
set. `empty_set()`

is equivalent to
`from_term([], ['_'])`

.

### extension(BinRel1, Set, AnySet) -> BinRel2

`AnySet = anyset()`

`BinRel1 = BinRel2 = binary_relation()`

`Set = 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 [, Type]) -> Family

`Family = family()`

`Tuples = [tuple()]`

`Type = type()`

Creates a family of subsets.
`family(F, T)`

is equivalent to
`from_term(F, T)`

, if the result is a family. If
no type is explicitly
given, `[{atom, [atom]}]`

is used as type of the
family.

### family_difference(Family1, Family2) -> Family3

`Family1 = Family2 = Family3 = family()`

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

1>`F1 = sofs:family([{a,[1,2]},{b,[3,4]}]),`

`F2 = sofs:family([{b,[4,5]},{c,[6,7]}]),`

`F3 = sofs:family_difference(F1, F2),`

`sofs:to_external(F3).`

[{a,[1,2]},{b,[3]}]

### family_domain(Family1) -> Family2

`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

`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

`Family1 = Family2 = family()`

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

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

message.

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

`F2 = sofs:family_intersection(F1),`

`sofs:to_external(F2).`

[{a,[2,3]},{b,[x,y]}]

### family_intersection(Family1, Family2) -> Family3

`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

`SetFun = set_fun()`

`Family1 = Family2 = family()`

`Set = set()`

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({sofs, union}, F1),`

`sofs:to_external(F2).`

[{a,[1,2,3]},{b,[]}]

### family_range(Family1) -> Family2

`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

`Fun = spec_fun()`

`Family1 = Family2 = family()`

If Family1 is a family,
then Family2 is
the restriction of
Family1 to those elements i of the
index set for which Fun applied to Family1[i] returns
`true`

. If Fun is a tuple `{external, Fun2}`

,
Fun2 is applied to
the external set of
Family1[i], otherwise Fun is applied to Family1[i].

1>`F1 = sofs:family([{a,[1,2,3]},{b,[1,2]},{c,[1]}]),`

`SpecFun = fun(S) -> sofs:no_elements(S) =:= 2 end,`

`F2 = sofs:family_specification(SpecFun, F1),`

`sofs:to_external(F2).`

[{b,[1,2]}]

### family_to_digraph(Family [, GraphType]) -> Graph

`Graph = digraph()`

`Family = family()`

`GraphType = - see digraph(3) -`

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

If no graph type is given, `digraph:new/1`

is used for
creating the directed graph, otherwise the GraphType
argument is passed on as second argument to
`digraph:new/2`

.

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

`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

`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({sofs,union}, F)`

.

### family_union(Family1, Family2) -> Family3

`Family1 = Family2 = Family3 = family()`

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

1>`F1 = sofs:family([{a,[1,2]},{b,[3,4]},{c,[5,6]}]),`

`F2 = sofs:family([{b,[4,5]},{c,[7,8]},{d,[9,10]}]),`

`F3 = sofs:family_union(F1, F2),`

`sofs:to_external(F3).`

[{a,[1,2]},{b,[3,4,5]},{c,[5,6,7,8]},{d,[9,10]}]

### field(BinRel) -> Set

`BinRel = binary_relation()`

`Set = 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

`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

`Set = set()`

`ListOfSets = [anyset()]`

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

1>`S1 = sofs:relation([{a,1},{b,2}]),`

`S2 = sofs:relation([{x,3},{y,4}]),`

`S = sofs:from_sets([S1,S2]),`

`sofs:to_external(S).`

[[{a,1},{b,2}],[{x,3},{y,4}]]

### from_sets(TupleOfSets) -> Ordset

`Ordset = ordset()`

`TupleOfSets = tuple-of(anyset())`

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

### from_term(Term [, Type]) -> AnySet

`AnySet = anyset()`

`Term = term()`

`Type = type()`

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

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

`sofs:to_external(S).`

[{{"foo"},[1]},{"foo",[2]}]

`from_term`

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

1>`A = sofs:from_term(a),`

`S = sofs:set([1,2,3]),`

`P1 = sofs:from_sets({A,S}),`

`P2 = sofs:from_term({b,[6,5,4]}),`

`Ss = sofs:from_sets([P1,P2]),`

`sofs:to_external(Ss).`

[{a,[1,2,3]},{b,[4,5,6]}]

Other functions that create sets are `from_external/2`

and `from_sets/1`

. Special cases of `from_term/2`

are `a_function/1,2`

, `empty_set/0`

,
`family/1,2`

, `relation/1,2`

, and `set/1,2`

.

### image(BinRel, Set1) -> Set2

`BinRel = binary_relation()`

`Set1 = Set2 = set()`

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

1>`R = sofs:relation([{1,a},{2,b},{2,c},{3,d}]),`

`S1 = sofs:set([1,2]),`

`S2 = sofs:image(R, S1),`

`sofs:to_external(S2).`

[a,b,c]

### intersection(SetOfSets) -> Set

`Set = 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

`Set1 = Set2 = Set3 = set()`

Returns the intersection of Set1 and Set2.

### intersection_of_family(Family) -> Set

`Family = family()`

`Set = set()`

Returns the intersection of the family Family.

Intersecting an empty family exits the process with a
`badarg`

message.

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

`S = sofs:intersection_of_family(F),`

`sofs:to_external(S).`

[2]

### inverse(Function1) -> Function2

`Function1 = Function2 = function()`

Returns the inverse of the function Function1.

1>`R1 = sofs:relation([{1,a},{2,b},{3,c}]),`

`R2 = sofs:inverse(R1),`

`sofs:to_external(R2).`

[{a,1},{b,2},{c,3}]

### inverse_image(BinRel, Set1) -> Set2

`BinRel = binary_relation()`

`Set1 = Set2 = 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

`Bool = bool()`

`BinRel = binary_relation()`

Returns `true`

if the binary relation BinRel is a
function or the
untyped empty set, `false`

otherwise.

### is_disjoint(Set1, Set2) -> Bool

`Bool = bool()`

`Set1 = Set2 = set()`

Returns `true`

if Set1 and Set2
are disjoint, `false`

otherwise.

### is_empty_set(AnySet) -> Bool

`AnySet = anyset()`

`Bool = bool()`

Returns `true`

if Set is an empty unordered set,
`false`

otherwise.

### is_equal(AnySet1, AnySet2) -> Bool

`AnySet1 = AnySet2 = anyset()`

`Bool = bool()`

Returns `true`

if the AnySet1 and AnySet2
are equal, `false`

otherwise. This example shows that `==/2`

is used when
comparing sets for equality:

1>`S1 = sofs:set([1.0]),`

`S2 = sofs:set([1]),`

`sofs:is_equal(S1, S2).`

true

### is_set(AnySet) -> Bool

`AnySet = anyset()`

`Bool = bool()`

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

`Bool = bool()`

`Term = term()`

Returns `true`

if Term is
an unordered set, an
ordered set or an atomic set, `false`

otherwise.

### is_subset(Set1, Set2) -> Bool

`Bool = bool()`

`Set1 = Set2 = set()`

Returns `true`

if Set1 is
a subset of Set2, `false`

otherwise.

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

`Relation1 = Relation2 = Relation3 = relation()`

`I = J = integer() > 0`

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

`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

`ASet = set() | ordset()`

`NoElements = integer() >= 0`

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

### partition(SetOfSets) -> Partition

`SetOfSets = set_of_sets()`

`Partition = 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

`SetFun = set_fun()`

`Partition = set()`

`Set = 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}

`SetFun = set_fun()`

`Set1 = Set2 = Set3 = Set4 = set()`

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

1>`R1 = sofs:relation([{1,a},{2,b},{3,c}]),`

`S = sofs:set([2,4,6]),`

`{R2,R3} = sofs:partition(1, R1, S),`

`{sofs:to_external(R2),sofs:to_external(R3)}.`

{[{2,b}],[{1,a},{3,c}]}

`partition(F, S1, S2)`

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

.

### partition_family(SetFun, Set) -> Family

`Family = family()`

`SetFun = set_fun()`

`Set = set()`

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

1>`S = sofs:relation([{a,a,a,a},{a,a,b,b},{a,b,b,b}]),`

`SetFun = {external, fun({A,_,C,_}) -> {A,C} end},`

`F = sofs:partition_family(SetFun, S),`

`sofs:to_external(F).`

[{{a,a},[{a,a,a,a}]},{{a,b},[{a,a,b,b},{a,b,b,b}]}]

### product(TupleOfSets) -> Relation

`Relation = relation()`

`TupleOfSets = tuple-of(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

`BinRel = binary_relation()`

`Set1 = Set2 = 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

`SetFun = set_fun()`

`Set1 = Set2 = 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

`BinRel = binary_relation()`

`Set = 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 [, Type]) -> Relation

`N = integer()`

`Type = N | type()`

`Relation = relation()`

`Tuples = [tuple()]`

Creates a relation.
`relation(R, T)`

is equivalent to
`from_term(R, T)`

, if T is
a type and the result is a
relation. If Type is an integer N, then
`[{atom, ..., atom}])`

, where the size of the
tuple is N, is used as type of the relation. If no type is
explicitly given, the size of the first tuple of Tuples is
used if there is such a tuple. `relation([])`

is
equivalent to `relation([], 2)`

.

### relation_to_family(BinRel) -> Family

`Family = family()`

`BinRel = binary_relation()`

### relative_product(TupleOfBinRels [, BinRel1]) -> BinRel2

`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 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}}]

Note that `relative_product({R1}, R2)`

is
different from `relative_product(R1, R2)`

; the
tuple of one element is not identified with the element
itself.

### relative_product(BinRel1, BinRel2) -> BinRel3

`BinRel1 = BinRel2 = BinRel3 = binary_relation()`

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

### relative_product1(BinRel1, BinRel2) -> BinRel3

`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

`BinRel1 = BinRel2 = binary_relation()`

`Set = 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

`SetFun = set_fun()`

`Set1 = Set2 = Set3 = set()`

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

1>`S1 = sofs:relation([{1,a},{2,b},{3,c}]),`

`S2 = sofs:set([b,c,d]),`

`S3 = sofs:restriction(2, S1, S2),`

`sofs:to_external(S3).`

[{2,b},{3,c}]

### set(Terms [, Type]) -> Set

`Set = set()`

`Terms = [term()]`

`Type = type()`

Creates an unordered
set. `set(L, T)`

is equivalent to
`from_term(L, T)`

, if the result is an unordered
set. If no type is
explicitly given, `[atom]`

is used as type of the set.

### specification(Fun, Set1) -> Set2

`Fun = spec_fun()`

`Set1 = Set2 = 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({sofs,is_a_function}, S1),`

`sofs:to_external(S2).`

[[{a,1},{b,2}]]

### strict_relation(BinRel1) -> BinRel2

`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

`SetFun = set_fun()`

`Set1 = Set2 = set()`

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

1>`L = [{a,1},{b,2}].`

[{a,1},{b,2}] 2>`sofs:to_external(sofs:projection(1,sofs:relation(L))).`

[a,b] 3>`sofs:to_external(sofs:substitution(1,sofs:relation(L))).`

[{{a,1},a},{{b,2},b}] 4>`SetFun = {external, fun({A,_}=E) -> {E,A} end},`

`sofs:to_external(sofs:projection(SetFun,sofs:relation(L))).`

[{{a,1},a},{{b,2},b}]

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

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

`sofs:to_external(I).`

[{a,a},{b,b},{c,c}]

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

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

Here might be the place to reveal something that was more
or less stated before, namely that external unordered sets
are represented as sorted lists. As a consequence, creating
the image of a set under a relation R may traverse all
elements of R (to that comes the sorting of results, the
image). In `images/2`

, BinRel will be traversed once
for each element of SetOfSets, which may take too long. The
following efficient function could be used instead under the
assumption that the image of each element of SetOfSets under
BinRel is non-empty:

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

### symdiff(Set1, Set2) -> Set3

`Set1 = Set2 = Set3 = 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}

`Set1 = Set2 = Set3 = Set4 = Set5 = 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

`ExternalSet = external_set()`

`AnySet = anyset()`

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

### to_sets(ASet) -> Sets

`ASet = set() | ordset()`

`Sets = tuple_of(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

`AnySet = anyset()`

`Type = type()`

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

### union(SetOfSets) -> Set

`Set = set()`

`SetOfSets = set_of_sets()`

Returns the union of the set of sets SetOfSets.

### union_of_family(Family) -> Set

`Family = family()`

`Set = set()`

Returns the union of the family Family.

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

`S = sofs:union_of_family(F),`

`sofs:to_external(S).`

[0,1,2,3,4]

### weak_relation(BinRel1) -> BinRel2

`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

- a_function/2
- canonical_relation/1
- composite/2
- constant_function/2
- converse/1
- difference/2
- digraph_to_family/2
- domain/1
- drestriction/2
- drestriction/3
- empty_set/0
- extension/3
- family/2
- family_difference/2
- family_domain/1
- family_field/1
- family_intersection/1
- family_intersection/2
- family_projection/2
- family_range/1
- family_specification/2
- family_to_digraph/2
- family_to_relation/1
- family_union/1
- family_union/2
- field/1
- from_external/2
- from_sets/1
- from_sets/1-1
- from_term/2
- image/2
- intersection/1
- intersection/2
- intersection_of_family/1
- inverse/1
- inverse_image/2
- is_a_function/1
- is_disjoint/2
- is_empty_set/1
- is_equal/2
- is_set/1
- is_sofs_set/1
- is_subset/2
- is_type/1
- join/4
- multiple_relative_product/2
- no_elements/1
- partition/1
- partition/2
- partition/3
- partition_family/2
- product/1
- product/2
- projection/2
- range/1
- relation/2
- relation_to_family/1
- relative_product/2
- relative_product/2-1
- relative_product1/2
- restriction/2
- restriction/3
- set/2
- specification/2
- strict_relation/1
- substitution/2
- symdiff/2
- symmetric_partition/2
- to_external/1
- to_sets/1
- type/1
- union/1
- union/2
- union_of_family/1
- weak_relation/1