sofs(3erl)

📦
version 25.3.2.8 · 5 versions
, in ubuntu noble

💅 sofs(3erl) — Erlang Module Definition 💅

✨ NAME ✨

sofs — Functions for manipulating sets of sets. That’s the whole assignment, bestie: sets of sets, and the functions that handle them. 🎀

🌸 DESCRIPTION 🌸

Okay so this module gives you operations on finite sets and relations that are represented as sets. Intuitively, a set is a collection of elements; every element belongs to the set, and the set contains every element. Yes, that is the vibe. ✨

The data that represents sofs is meant to be treated as opaque by other modules. In abstract terms, the representation is a composite type of existing Erlang terms. See note on data types. Any code assuming knowledge of the format is running on thin ice. 💀

Given a set A and a sentence S(x), where x is a free variable, you can form a new set B whose elements are exactly those elements of A for which S(x) holds. That’s 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 — which is always the case in this module — it’s denoted B = {x : S(x)}. No cap, that’s the foundation. 🧚

  • 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. Periodt. 💅

  • 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 elements 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 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). 🫶

So the types are literally how sets get their vibe check ✅ — they encode the conditions a set has to satisfy. Take the relative product of two sets R and S (iykyk): it's defined when R is a binary relation to Y and S is a binary relation from Y. The function that does the deed, relative_product/2, verifies both args are binary relations by matching [{A,B}] against the type of the first argument (let's call it Arg1) and [{C,D}] against the type of the second (Arg2). When [{A,B}] matches the type of Arg1, read that as "Arg1 represents a binary relation from X to Y," where X is defined as all sets Set(x) for some element x in Sets whose type is A — and same deal for Y. Identically, Arg2 is read as a binary relation from W to Z. Then it checks B matches C, which is enough to guarantee W equals Y. The untyped empty set is handled on its own little track: its type, ['_'], matches the type of any unordered set 💫

A handful of functions in this module (drestriction/3, family_projection/2, partition/2, partition_family/2, projection/2, restriction/3, substitution/2) take an Erlang function as a way to transform each element of a given unordered set. That function — called SetFun from here on — can be given as a functional object (fun), a tuple {external, Fun}, or an integer:

  • If SetFun is a fun, the fun gets applied to each element of the given set, and whatever comes back is assumed to be a set.
  • If SetFun is the tuple {external, Fun}, then Fun gets applied to the external set of each element of the given set, and the return value is assumed to be an external set. Picking out the elements of an unordered set as external sets and assembling a fresh unordered set from a list of external sets is, in this implementation, more efficient than modifying each element as a set. But bestie, this optimization only works when the elements of the unordered set are atomic or ordered sets. Also required: the type of the elements matches some clause of Fun (the type of the created set is whatever you get from applying Fun to the type of the given set), and Fun does nothing except select, duplicate, or rearrange parts of the elements. No funny business 🔍
  • Specifying SetFun as an integer I is equivalent to specifying {external, fun(X) -> element(I, X) end}, but the integer is the preferred glow-up — it makes handling this case even more efficient ✨

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 it can totally change in future versions of this module. So don't build your whole personality around it 🙈

The execution time of this module's functions is dominated by how long it takes to sort lists. When no sorting is needed, worst case runtime is proportional to the sum of the sizes of the input arguments and the returned value. A few functions are constant time, and they're iconic for it: 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 you hand them badly formed arguments or sets whose types don't get along. Not a vibe.

When comparing external sets, operator ==/2 is used.

🦋 DATA TYPES 🦋

  • anyset() = ordset() | a_set() — any kind of set (atomic sets are also part of the club).
  • 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 you don't explicitly specify a type, [{atom, atom}] is used as the function type. Defaults, slay 💖

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. Canonical queen behavior 👑

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. Two functions, one glow-up ✨

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. Everyone gets the same answer, no cap 💀

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. Flip it and reverse it 🦋

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 the type of the family. It is assumed that Type is a valid type of the external set of the family. Careful with that assumption bestie 🫶

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)). Round trip, no losses 💫

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. It's the restriction's evil twin 😈

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. Filtering, but make it fancy 🌸

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([], ['_']). Nothing in it, but make it iconic 🎀

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). Filling in the blanks, bestie 🫶

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. Same energy as a_function, different flavor 💜

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]. Per-index subtraction, no notes 🔥

okay so bestie, picking up right where we left off with the family_difference example — the rest is literally this:

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()

So if Family1 is a family and every Family1[i] is a binary relation, then Family2 is the family with the same index set as Family1, except each Family2[i] is the domain of Family1[i]. domain girlie pop ✨

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, then Family2 keeps the same index set as Family1, and Family2[i] is the field of Family1[i]. field trip energy 🦋

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)). no cap, it's giving the same result 💅

family_intersection(Family1) -> Family2

Types:

  • Family1 = Family2 = family()

If Family1 is a family and every Family1[i] is a set of sets, then Family2 has the same index set as Family1, and each Family2[i] is the intersection of Family1[i]. intersection slay 🎀

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's index set is the intersection of the index sets of Family1 and Family2, and Family3[i] is the intersection of Family1[i] and Family2[i]. overlap era 💖

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 has the same index set as Family1, and Family2[i] is what you get when you call SetFun with Family1[i] as the argument. projection pop off 🔥

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, then Family2 has the same index set as Family1, and Family2[i] is the range of Family1[i]. range baddie 🌸

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]. filtered and fabulous 🧚

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 the 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. graph goblin mode 👑

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.

If 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]. relation retrograde 🫶

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 has the same index set as Family1, and Family2[i] is the union of Family1[i]. union unite 🌈

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). iykyk 💅

family_union(Family1, Family2) -> Family3

Types:

  • Family1 = Family2 = Family3 = family()

If Family1 and Family2 are families, then Family3's index set is the union of the index sets of Family1 and Family2, and Family3[i] is the union of Family1[i] and Family2[i] if both map i, otherwise Family1[i] or Family2[i]. merged and majestic 💖

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. field notes 🍓

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)). periodt ✨

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. external but make it cute 🎀

from_sets(ListOfSets) -> Set

Types:

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

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

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. ordered and unbothered ⭐

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 external set you end up with. 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: traversal bestie 🔥

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": layer by layer realness 🌸

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. set creation station 🌈

image(BinRel, Set1) -> Set2

Types:

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

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

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. intersection initiation 🎀

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. two-set tango 💖

💖 intersection_of_family(Family) -> Set

Types:

Family = family()
Set = a_set()

Okay so this one gives you the intersection of the family Family — like, the overlap era of every set in the fam. Slay.

🚨 Heads up bestie: intersecting an empty family exits the process with a badarg message. It does not vibe check first, it just ends things. 💀
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. Flip era. 🔄

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. It’s giving “who maps into this set?” energy. 💅

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. Periodt.

✨ is_disjoint(Set1, Set2) -> Bool

Types:

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

Returns true if Set1 and Set2 are disjoint, otherwise false. No overlap, no drama. 🫶

🧚 is_empty_set(AnySet) -> Bool

Types:

AnySet = anyset()
Bool = boolean()

Returns true if AnySet is an empty unordered set, otherwise false. That’s it, that’s the function. 💖

💜 is_equal(AnySet1, AnySet2) -> Bool

Types:

AnySet1 = AnySet2 = anyset()
Bool = boolean()

Returns true if AnySet1 and AnySet2 are equal, otherwise false.

And bestie, the comparison uses ==/2 when checking equality. The example is literally there to show you that:

1> S1 = sofs:set([1.0]),
S2 = sofs:set([1]),
sofs:is_equal(S1, S2).
true

So yeah, 1.0 and 1 are matching in this context. Iconic. 🌈

👑 is_set(AnySet) -> Bool

Types:

AnySet = anyset()
Bool = boolean()

Returns true if AnySet appears to be an unordered set, and false if AnySet is an ordered set or an atomic set or any other term.

🫶 Important note: the test is shallow, and this function will return true for any term that coincides with the representation of an unordered set. See also the note on data types. So like, it’s not doing a deep vibe check, okay?

⭐ is_sofs_set(Term) -> Bool

Types:

Bool = boolean()
Term = term()

Returns true if Term appears to be an unordered set, an ordered set, or an atomic set, otherwise false.

🌸 Again: this function will return true for any term that coincides with the representation of a sofs set. See also the note on data types. Shallow test stays shallow, bestie.

🍓 is_subset(Set1, Set2) -> Bool

Types:

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

Returns true if Set1 is a subset of Set2, otherwise false. Tiny set in a bigger set? Slay. 💅

🧚‍♀️ is_type(Term) -> Bool

Types:

Bool = boolean()
Term = term()

Returns true if term Term is a type. That’s the whole thing. ✨

🔥 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. It’s a matching moment, no cap.

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. Count ‘em, bestie. 🔢

💖 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. It’s a grouping era. 🎀

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. Same result, same group. Easy. 💅

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}]}

And just so you know: partition(F, S1, S2) is equivalent to {restriction(F, S1, S2), drestriction(F, S1, S2)}. Exact same energy. ✨

🧚 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. Very main character indexing behavior. 👑

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. It’s giving “every combination, every slay.” 🔥

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. Pairing era. 💖

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}]

Also, product(S1, S2) is equivalent to product({S1, S2}). Same vibe, different font. ✨

🦋 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. Glow-up transformation, basically. 💅

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. All the outputs, bestie. 🫶

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). So yeah, defaults are doing their thing. ✨

💜 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. Domain-to-family pipeline, slay. 👑

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. Chained relation energy, no cap. 🔥

Okay so 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(3erl), digraph(3erl), orddict(3erl), ordsets(3erl), sets(3erl) — iconic little crossover episode, bestie 💅

Cute sign-off: Ericsson AB, stdlib 4.3.1.3, sofs(3erl) — that’s the source/version/date footer vibe, and we’re leaving it cute instead of copy-pasting the line 🎀💖

📦 where this page comes from