## table of contents

- stretch 1:19.2.1+dfsg-2+deb9u2
- testing 1:21.2.6+dfsg-1
- unstable 1:21.2.6+dfsg-1
- experimental 1:22.0.4+dfsg-1

sofs(3erl) | Erlang Module Definition | sofs(3erl) |

# NAME¶

sofs - Functions for manipulating sets of sets.# DESCRIPTION¶

This module provides operations on finite sets and relations represented as sets. Intuitively, a set is a collection of elements; every element belongs to the set, and the set contains every element.Given a set A and a sentence S(x), where x is a free variable, a new set B whose elements are exactly those elements of A for which S(x) holds can be formed, this is denoted B = {x in A : S(x)}. Sentences are expressed using the logical operators "for some" (or "there exists"), "for all", "and", "or", "not". If the existence of a set containing all the specified elements is known (as is always the case in this module), this is denoted B = {x : S(x)}.

- *
- 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 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 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 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 ... x 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] x ... x
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 ... x 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/2* creates 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/2* with S1 and S2 as arguments, then U is said to
be the union of S1 and S2. A more precise formulation is that Set(U) is the
union of Set(S1) and Set(S2).

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

A few functions of this module (*drestriction/3*,
*family_projection/2*, *partition/2*,
*partition_family/2*, *projection/2*,
*restriction/3*, *substitution/2*) accept an Erlang
function as a means to modify each element of a given unordered set. Such a
function, called SetFun in the following, can be specified as a functional
object (fun), a tuple *{external, Fun}*, or an integer:

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

Examples of SetFuns:

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

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

The execution time of the functions of this module is dominated by
the time it takes to sort lists. When no sorting is needed, the execution
time is in the worst case proportional to the sum of the sizes of the input
arguments and the returned value. A few functions execute in constant time:
*from_external/2*, *is_empty_set/1*,
*is_set/1*, *is_sofs_set/1*,
*to_external/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:

**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:

**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:

**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()**

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:

**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:

**a_set()**

Returns the **difference** of the sets *Set1* and
*Set2*.

digraph_to_family(Graph) -> Family

digraph_to_family(Graph, Type) -> Family

Types:

**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:

**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:

**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:

**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:

**a_set()**

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

extension(BinRel1, Set, AnySet) -> BinRel2

Types:

**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()**

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:

**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:

**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:

**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:

**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:

**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:

**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:

**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:

**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:

**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()**

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:

**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:

**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:

**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:

**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:

**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()**

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()**

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:

**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:

**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:

**a_set()**

Returns the **intersection** of *Set1* and
*Set2*.

intersection_of_family(Family) -> Set

Types:

**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:

**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:

**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:

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:

Set1 = Set2 =

**a_set()**

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

is_empty_set(AnySet) -> Bool

Types:

**anyset()**

Bool = boolean()

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

is_equal(AnySet1, AnySet2) -> Bool

Types:

**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()**

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:

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:

Set1 = Set2 =

**a_set()**

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

is_type(Term) -> Bool

Types:

Term = term()

Returns *true* if term *Term* is a **type**.

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

Types:

**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:

**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:

**a_set()**|

**ordset()**

NoElements = integer() >= 0

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

partition(SetOfSets) -> Partition

Types:

**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:

**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:

**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()**

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()**

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:

**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:

**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:

**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:

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()**

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:

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:

**binary_relation()**

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

relative_product1(BinRel1, BinRel2) -> BinRel3

Types:

**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:

**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:

**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:

**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:

**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:

**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:

**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:

**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:

**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:

**external_set()**

AnySet =

**anyset()**

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

to_sets(ASet) -> Sets

Types:

**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()**

Type =

**type()**

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

union(SetOfSets) -> Set

Types:

**a_set()**

SetOfSets =

**set_of_sets()**

Returns the **union** of the set of sets *SetOfSets*.

union(Set1, Set2) -> Set3

Types:

**a_set()**

Returns the **union** of *Set1* and *Set2*.

union_of_family(Family) -> Set

Types:

**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:

**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(3erl)*,

*digraph(3erl)*,

*orddict(3erl)*,

*ordsets(3erl)*,

*sets(3erl)*

stdlib 3.2 | Ericsson AB |