From 68d53c01b0b8e9a007a6a30158c19e34b2d2a34e Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Bj=C3=B6rn=20Gustavsson?= Date: Wed, 18 May 2016 15:53:35 +0200 Subject: Update STDLIB documentation MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Language cleaned up by the technical writers xsipewe and tmanevik from Combitech. Proofreading and corrections by Björn Gustavsson and Hans Bolinder. --- lib/stdlib/doc/src/sofs.xml | 1345 ++++++++++++++++++++++++------------------- 1 file changed, 740 insertions(+), 605 deletions(-) (limited to 'lib/stdlib/doc/src/sofs.xml') diff --git a/lib/stdlib/doc/src/sofs.xml b/lib/stdlib/doc/src/sofs.xml index 1e5be367bd..4cf1984d46 100644 --- a/lib/stdlib/doc/src/sofs.xml +++ b/lib/stdlib/doc/src/sofs.xml @@ -24,260 +24,284 @@ sofs Hans Bolinder - nobody + - nobody - no + + 2001-08-25 PA1 - sofs.sgml + sofs.xml sofs - Functions for Manipulating Sets of Sets + Functions for manipulating sets of sets. -

The sofs module implements operations on finite sets and +

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 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:

+ specified elements is known (as is always the case in this + module), this is denoted B = {x : S(x)}.

+ - for every atom T except '_' and for every term X, - (T, X) belongs to Sets (atomic sets); + +

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.

 - (['_'], []) belongs to Sets (the untyped empty 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.

 - 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); + +

The Cartesian + product of two sets X and Y, denoted X × Y, is + the set {a : a = (x, y) for some x in X and + for some y in Y}.

+

A relation is a subset of + X × Y. Let R be a relation. The fact that (x, y) + belongs to R is written as x R y. As relations are sets, + the definitions of the last item (subset, union, and so on) apply to + relations as well.

+

The domain of R is the set + {x : x R y for some y in Y}.

+

The range of R is the set + {y : x R y for some x in X}.

+

The converse of R is the + set {a : a = (y, x) for some + (x, y) in R}.

+

If A is a subset of X, the image + of A under R is the set {y : x R y for some + x in A}. If B is a subset of Y, the + inverse image of B is the + set {x : x R y for some y in B}.

+

If R is a relation from X to Y, and S is a relation from Y to Z, the + relative product of R + and S is the relation T from X to Z defined so that x T z + if and only if there exists an element y in Y such that + x R y and y S z.

+

The restriction of R to A + is the set S defined so that x S y if and only if there + exists an element x in A such that x R y.

+

If S is a restriction of R to A, then R is an + extension of S to X.

+

If X = Y, then R is called a relation in X.

+

The field of a relation R in X + is the union of the domain of R and the range of R.

+

If R is a relation in X, and if S is defined so that x S y + if x R y and not x = y, then S is the + strict relation + corresponding to R. Conversely, if S is a relation in X, and if R is + defined so that x R y if x S y or x = y, + then R is the weak + relation corresponding to S.

+

A relation R in X is reflexive if x R x for every + element x of X, it is symmetric if x R y implies + that y R x, and it is transitive if + x R y and y R z imply that x R z.

+ + +

A function F is a relation, + a subset of X × Y, such that the domain of F is equal + to X and such that for every x in X there is a unique element y in Y + with (x, y) in F. The latter condition can be formulated as + follows: if x F y and x F z, then y = z. + In this module, it is not required that the domain of F is equal to X + for a relation to be considered a function.

+

Instead of writing (x, y) in F or x F y, we + write F(x) = y when F is a function, and say that F maps x + onto y, or that the value of F at x is y.

+

As functions are relations, the definitions of the last item (domain, + range, and so on) apply to functions as well.

+

If the converse of a function F is a function F', then F' is called + the inverse of F.

+

The relative product of two functions F1 and F2 is called + the composite of F1 and F2 + if the range of F1 is a subset of the domain of F2.

+ + +

Sometimes, when the range of a function is more important than the + function itself, the function is called a family.

+

The domain of a family is called the index set, and the + range is called the indexed set.

+

If x is a family from I to X, then x[i] denotes the value of the + function at index i. The notation "a family in X" is used for such a + family.

+

When the indexed set is a set of subsets of a set X, we call x a + family of subsets of X.

+

If x is a family of subsets of X, the union of the range of x is + called the union of the family x.

+

If x is non-empty (the index set is non-empty), the intersection + of the family x is the intersection of the range of x.

+

In this module, the only families that are considered are families + of subsets of some set X; in the following, the word "family" is + used for such families of subsets.

+ + +

A partition of a set X is a + collection S of non-empty subsets of X whose union is X and whose + elements are pairwise disjoint.

+

A relation in a set is an equivalence relation if it is + reflexive, symmetric, and transitive.

+

If R is an equivalence relation in X, and x is an element of X, the + equivalence class of + x with respect to R is the set of all those elements y of X for which + x R y holds. The equivalence classes constitute a + partitioning of X. Conversely, if C is a partition of X, the relation + that holds for any two elements of X if they belong to the same + equivalence class, is an equivalence relation induced by the + partition C.

+

If R is an equivalence relation in X, the + canonical map is the + function that maps every element of X onto its equivalence class.

+ + +

Relations as defined above + (as sets of ordered pairs) are from now on referred to as binary + relations.

+

We call a set of ordered sets (x[1], ..., x[n]) an + (n-ary) relation, and + say that the relation is a subset of the + Cartesian product + X[1] × ... × X[n], where x[i] is + an element of X[i], 1 <= i <= n.

+

The projection of an n-ary + relation R onto coordinate i is the set {x[i] : + (x[1], ..., x[i], ..., x[n]) in R for some + x[j] in X[j], 1 <= j <= n and + not i = j}. The projections of a binary relation R onto the + first and second coordinates are the domain and the range of R, + respectively.

+

The relative product of binary relations can be generalized to n-ary + relations as follows. Let TR be an ordered set + (R[1], ..., R[n]) of binary relations from X to Y[i] + and S a binary relation from + (Y[1] × ... × Y[n]) to Z. The + relative product + of TR and S is the binary relation T from X to Z defined so that + x T z if and only if there exists an element y[i] in Y[i] + for each 1 <= i <= n such that + x R[i] y[i] and + (y[1], ..., y[n]) S z. Now let TR be a an + ordered set (R[1], ..., R[n]) of binary relations from + X[i] to Y[i] and S a subset of + X[1] × ... × X[n]. + The multiple + relative product of TR and S is defined to be the set + {z : z = ((x[1], ..., x[n]), (y[1],...,y[n])) + for some (x[1], ..., x[n]) in S and for some + (x[i], y[i]) in R[i], 1 <= i <= n}.

+

The natural join of an + n-ary relation R and an m-ary relation S on coordinate i and j is + defined to be the set + {z : z = (x[1], ..., x[n],  + y[1], ..., y[j-1], y[j+1], ..., y[m]) + for some (x[1], ..., x[n]) in R and for some + (y[1], ..., y[m]) in S such that + x[i] = y[j]}.

+ + +

The sets recognized by this + module are represented by elements of the relation Sets, which is + defined as the smallest set such that:

+ + +

For every atom T, except '_', and for every term X, + (T, X) belongs to Sets (atomic sets).

+ + +

(['_'], []) belongs to Sets (the untyped empty + set).

+ + +

For every tuple T = {T[1], ..., T[n]} and + for every tuple X = {X[1], ..., X[n]}, if + (T[i], X[i]) belongs to Sets for every + 1 <= i <= n, then (T, X) belongs + to Sets (ordered sets).

+ + +

For every term T, if X is the empty list or a non-empty + sorted list [X[1], ..., X[n]] without duplicates + such that (T, X[i]) belongs to Sets for every + 1 <= i <= n, then ([T], X) + belongs to Sets (typed unordered sets).

+ + +

An external set is an + element of the range of Sets.

+

A type is an element of the + domain of Sets.

+

If S is an element (T, X) of Sets, then T is a + valid type of X, T is the + type of S, and X is the external set of S. + from_term/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).

 - 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 + 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 + 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 @@ -290,33 +314,51 @@ 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 + +

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:

+ 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
@@ -325,22 +367,31 @@ fun(S) -> sofs:partition(1, S) 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.

+ 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, - is_empty_set, is_set, is_sofs_set, - to_external, type.

+ 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 the operator ==/2 is used.

+ +

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

 + @@ -402,6 +453,7 @@ fun(S) -> sofs:partition(1, S) end

A tuple where the elements are of type T.

 + @@ -410,24 +462,25 @@ fun(S) -> sofs:partition(1, S) end

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

+ specified, [{atom, atom}] is used as the + function type.

 + Return the canonical map.

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.

+ (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),
@@ -435,13 +488,14 @@ fun(S) -> sofs:partition(1, S) end
 [{a,[a,b]},{b,[a,b]},{b,[b,c]},{c,[b,c]}]
 + Return the composite of two functions.

Returns the composite of - the functions Function1 and - Function2.

+ 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}]),
@@ -450,13 +504,14 @@ fun(S) -> sofs:partition(1, S) end
 [{a,x},{b,y},{c,y}]
 + - Create the function that maps each element of a + Create the function that maps each element of a set onto another set.

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

+ that maps each element of set Set onto AnySet.

 1> S = sofs:set([a,b]),
 E = sofs:from_term(1),
@@ -465,12 +520,13 @@ fun(S) -> sofs:partition(1, S) end
 [{a,1},{b,1}]
 + Return the converse of a binary relation.

Returns the converse - of the binary relation BinRel1.

+ of the binary relation BinRel1.

 1> R1 = sofs:relation([{1,a},{2,b},{3,a}]),
 R2 = sofs:converse(R1),
@@ -478,39 +534,42 @@ fun(S) -> sofs:partition(1, S) end
 [{a,1},{a,3},{b,2}]
 + Return the difference of two sets. -

Returns the difference of - the sets Set1 and Set2.

+

Returns the difference of + the sets Set1 and Set2.

 + Create a family from a directed graph.

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.

+ 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)).

 + Return the domain of a binary relation. -

Returns the domain of - the binary relation BinRel.

+

Returns the domain of + the binary relation BinRel.

 1> R = sofs:relation([{1,a},{1,b},{2,b},{2,c}]),
 S = sofs:domain(R),
@@ -518,14 +577,15 @@ fun(S) -> sofs:partition(1, S) end
 [1,2]
 + Return a restriction of a binary relation.

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

+ of BinRel1 to Set.

 1> R1 = sofs:relation([{1,a},{2,b},{3,c}]),
 S = sofs:set([2,4,6]),
@@ -536,14 +596,15 @@ fun(S) -> sofs:partition(1, S) end
           difference(R, restriction(R, S)).

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

+        
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}]),
@@ -555,24 +616,27 @@ fun(S) -> sofs:partition(1, S) end
           difference(S1, restriction(F, S1, S2)).

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

       
     
+
     
       
       Extend the domain of a binary relation.
       
-        
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).

+        
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(),
@@ -582,31 +646,33 @@ fun(S) -> sofs:partition(1, S) end
 [{a,[1,2]},{b,[3]},{c,[]}]

       
     
+
     
       
       
       Create a family of subsets.
       
-        
Creates a family of subsets. 
-	  family(F, T) is equivalent to
-          from_term(F, T), if the result is a family. If
+        
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.

+          specified, [{atom, [atom]}] is used as the
+          family type.

       
     
+
     
       
       Return the difference of two families.
       
-        
If Family1 and Family2
-	  are families, then 
-	  Family3 is the 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.

+          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]}]),
@@ -615,19 +681,20 @@ fun(S) -> sofs:partition(1, S) end
 [{a,[1,2]},{b,[3]}]

       
     
+
     
       
       Return a family of domains.
       
-        
If Family1 is
+        
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
+          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].

+          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),
@@ -635,43 +702,46 @@ fun(S) -> sofs:partition(1, S) end
 [{a,[1,2,3]},{b,[]},{c,[4,5]}]

       
     
+
     
       
       Return a family of fields.
       
-        
If Family1 is
+        
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
+          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].

+          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_union(family_domain(Family1),
+          family_range(Family1)).

       
     
+
     
       
       Return the intersection of a family
          of sets of sets.
       
-        
If Family1 is
+        
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
+          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
+          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]]}]),
@@ -680,17 +750,18 @@ fun(S) -> sofs:partition(1, S) end
 [{a,[2,3]},{b,[x,y]}]

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

+        
If Family1 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]}]),
@@ -699,17 +770,18 @@ fun(S) -> sofs:partition(1, S) end
 [{b,[4]},{c,[]}]

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

+        
If Family1 is
+          a family,
+          then Family2 is the 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),
@@ -717,19 +789,20 @@ fun(S) -> sofs:partition(1, S) end
 [{a,[1,2,3]},{b,[]}]

       
     
+
     
       
       Return a family of ranges.
       
-        
If Family1 is
+        
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
+          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].

+          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),
@@ -737,22 +810,23 @@ fun(S) -> sofs:partition(1, S) end
 [{a,[a,b,c]},{b,[]},{c,[d,e]}]

       
     
+
     
       
       Select a subset of a family using a predicate.
       
-        
If Family1 is
+        
If Family1 is
           a family,
-          then Family2 is
+          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
+          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].

+          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,
@@ -761,23 +835,24 @@ fun(S) -> sofs:partition(1, S) end
 [{b,[1,2]}]

       
     
+
     
       
       
       Create a directed graph from a family.
       
-        
Creates a directed graph from 
-	  the family Family.
+        
Creates a directed graph from
+          family Family.
           For each pair (a, {b[1], ..., b[n]})
-          of Family, the vertex
-          a as well the edges (a, b[i]) for
+          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 given 
-          digraph:new/0 is used for
-          creating the directed graph, otherwise the GraphType
-          argument is passed on as second argument to
-          digraph:new/1.

+        
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
@@ -786,16 +861,17 @@ fun(S) -> sofs:partition(1, S) end
           a cyclic message.

       
     
+
     
       
       Create a binary relation from a family.
       
-        
If Family is
+        
If Family is
           a family,
-          then BinRel is the binary relation containing
+          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].

+          of Family and x belongs
+          to Family[i].

         
 1> F = sofs:family([{a,[]}, {b,[1]}, {c,[2,3]}]),
 R = sofs:family_to_relation(F),
@@ -803,19 +879,20 @@ fun(S) -> sofs:partition(1, S) end
 [{b,1},{c,2},{c,3}]

       
     
+
     
       
       Return the union of a family of sets of sets.
       
-        
If Family1 is
+        
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
+          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].

+          Family1[i].

         
 1> F1 = sofs:from_term([{a,[[1,2],[2,3]]},{b,[[]]}]),
 F2 = sofs:family_union(F1),
@@ -825,19 +902,20 @@ fun(S) -> sofs:partition(1, S) end
           family_projection(fun sofs:union/1, F).

       
     
+
     
       
       Return the union of two families.
       
-        
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.

+        
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]}]),
@@ -846,40 +924,43 @@ fun(S) -> sofs:partition(1, S) end
 [{a,[1,2]},{b,[3,4,5]},{c,[5,6,7,8]},{d,[9,10]}]

       
     
+
     
       
       Return the field of a binary relation.
       
         
Returns the field of the
-          binary relation BinRel.

+          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)).

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

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

+          type of ExternalSet.

       
     
+
     
       
       Create a set out of a list of sets.
       
-        
Returns the unordered 
-          set containing the sets of the list
-          ListOfSets.

+        
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}]),
@@ -888,31 +969,33 @@ fun(S) -> sofs:partition(1, S) end
 [[{a,1},{b,2}],[{x,3},{y,4}]]

       
     
+
     
       
       Create an ordered set out of a tuple of sets.
       
-        
Returns the ordered 
-	  set containing the sets of the non-empty tuple 
-	  TupleOfSets.

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

       
     
+
     
       
       
       Create a set.
       
-        
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
+        
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 demonstrated by this example
-          where "foo" and {"foo"} are left unmodified:

+          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]}]),
@@ -920,12 +1003,12 @@ fun(S) -> sofs:partition(1, S) end
 [{{"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
+          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
-          may be the way to go if the ordered sets are big and one
+          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. An example showing that a set can be
+          the unordered set. The following example shows that a set can be
           built "layer by layer":

         
 1> A = sofs:from_term(a),
@@ -935,19 +1018,25 @@ fun(S) -> sofs:partition(1, S) end
 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.

+        
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.

       
     
+
     
       
       Return the image of a set under a binary relation.
       
-        
Returns the image of the
-          set Set1 under the binary
-          relation BinRel.

+        
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]),
@@ -956,32 +1045,35 @@ fun(S) -> sofs:partition(1, S) end
 [a,b,c]

       
     
+
     
       
       Return the intersection of a set of sets.
       
-        
Returns 
-	  the intersection of 
-	  the set of sets SetOfSets.

+        
Returns
+          the intersection of
+          the set of sets SetOfSets.

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

       
     
+
     
       
       Return the intersection of two sets.
       
-        
Returns 
-	  the intersection of 
-	  Set1 and Set2.

+        
Returns
+          the intersection of
+          Set1 and Set2.

       
     
+
     
       
       Return the intersection of a family.
       
-        
Returns the intersection of 
-	  the family Family.
+        
Returns the intersection of
+          family Family.
         

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

@@ -992,12 +1084,13 @@ fun(S) -> sofs:partition(1, S) end
 [2]

       
     
+
     
       
       Return the inverse of a function.
       
         
Returns the inverse
-          of the function Function1.

+          of function Function1.

         
 1> R1 = sofs:relation([{1,a},{2,b},{3,c}]),
 R2 = sofs:inverse(R1),
@@ -1005,14 +1098,15 @@ fun(S) -> sofs:partition(1, S) end
 [{a,1},{b,2},{c,3}]

       
     
+
     
       
-      Return the inverse image of a set under 
+      Return the inverse image of a set under
         a binary relation.
       
         
Returns the inverse
-	  image of Set1 under the binary
-	  relation BinRel.

+          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]),
@@ -1021,42 +1115,46 @@ fun(S) -> sofs:partition(1, S) end
 [2,3]

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

+          untyped empty set, otherwise false.

       
     
+
     
       
       Test for disjoint sets.
       
-        
Returns true if Set1
-	  and Set2
-	  are disjoint, false
-          otherwise.

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

       
     
+
     
       
       Test for an empty set.
       
-        
Returns true if AnySet is an empty
-          unordered set, false otherwise.

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

       
     
+
     
       
       Test two sets for equality.
       
-        
Returns true if the AnySet1
-	  and AnySet2
-	  are equal, false 
-	  otherwise. This example shows that ==/2 is used when
-          comparing sets for equality:

+        
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]),
@@ -1064,50 +1162,55 @@ fun(S) -> sofs:partition(1, S) end
 true

       
     
+
     
       
       Test for an unordered set.
       
-        
Returns true if AnySet is
-	  an unordered set, and
-          false if AnySet is an ordered set or an
+        
Returns true if AnySet is
+          an unordered set, and
+          false if AnySet is an ordered set or an
           atomic set.

       
     
+
     
       
       Test for an unordered set.
       
-        
Returns true if Term is
+        
Returns true if Term is
           an unordered set, an
-          ordered set or an atomic set, false otherwise.

+          ordered set, or an atomic set, otherwise false.

       
     
+
     
       
       Test two sets for subset.
       
-        
Returns true if Set1 is
-	  a subset
-	  of Set2, false otherwise.

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

       
     
+
     
       
       Test for a type.
       
-        
Returns true if the term Term is
-	  a type.

+        
Returns true if term Term is
+          a type.

       
     
+
     
       
       Return the join of two relations.
       
-        
Returns the natural 
-	  join of the relations Relation1
-	  and Relation2 on coordinates I and
-	  J.

+        
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}]),
@@ -1116,18 +1219,19 @@ true

 [{a,x,1,f,g},{a,x,1,h,i},{b,y,2,3,4}]

       
     
+
     
       
-      Return the multiple relative product of a tuple of binary 
+      Return the multiple relative product of a tuple of binary
         relations and a relation.
       
-        
If TupleOfBinRels is a non-empty tuple
+        
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.

+          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}]),
@@ -1136,22 +1240,24 @@ true

 [{1,2},{2,3},{3,1}]
 + Return the number of elements of a set.

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

+ set ASet.

 + Return the coarsest partition given a set of sets. -

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

+

Returns the partition of + 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]]),
@@ -1160,13 +1266,14 @@ true
[[a],[b,c],[d],[e,f],[g],[h,i],[j]] + Return a partition of a set. -

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

+

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,
@@ -1175,17 +1282,18 @@ true
[[[a],[b]],[[c,d],[e,f]]] + Return a partition of 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 yields an element in Set2, - the element belongs to Set3, otherwise the - element belongs to Set4.

+ 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]),
@@ -1193,23 +1301,23 @@ true
{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), + {restriction(F, S1, S2), drestriction(F, S1, S2)}.

 + Return a family indexing a partition. -

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.

+

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},
@@ -1218,16 +1326,16 @@ true
[{{a,a},[{a,a,a,a}]},{{a,b},[{a,a,b,b},{a,b,b,b}]}] + Return the Cartesian product of a tuple of sets. -

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

+

Returns the Cartesian + product of 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]),
@@ -1237,13 +1345,14 @@ true
[{a,1,x},{a,1,y},{a,2,x},{a,2,y},{b,1,x},{b,1,y},{b,2,x},{b,2,y}] + Return the Cartesian product of two sets. -

Returns the Cartesian - product of Set1 - and Set2.

+

Returns the Cartesian + product of Set1 + and Set2.

 1> S1 = sofs:set([1,2]),
 S2 = sofs:set([a,b]),
@@ -1254,17 +1363,18 @@ true
product({S1, S2}).

 + Return a set of substituted elements.

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 + 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.

+ Set1 onto coordinate i.

 1> S1 = sofs:from_term([{1,a},{2,b},{3,a}]),
 S2 = sofs:projection(2, S1),
@@ -1272,12 +1382,13 @@ true
[a,b] + Return the range of a binary relation.

Returns the range of the - binary relation BinRel.

+ binary relation BinRel.

 1> R = sofs:relation([{1,a},{1,b},{2,b},{2,c}]),
 S = sofs:range(R),
@@ -1285,6 +1396,7 @@ true
[a,b,c] + @@ -1292,27 +1404,28 @@ true

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 + 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).

 + Create a family from a binary relation. -

Returns the 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.

+

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),
@@ -1320,20 +1433,21 @@ true
[{b,[1]},{c,[2,3]}] + Return the relative product of a list of binary relations - and a binary relation. + and a binary relation. -

If ListOfBinRels is a non-empty list +

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 + 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 + BinRel1.

+

If BinRel1 is omitted, the relation of equality between the elements of the Cartesian product of the ranges of R[i], @@ -1345,33 +1459,33 @@ true 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 +

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.

+ list of one element is not identified with the element itself.

 + - Return the relative product of + Return the relative product of two binary relations. -

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

+

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

 + - Return the relative_product of + Return the relative_product of two binary relations. -

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

+

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}]),
@@ -1382,13 +1496,14 @@ true
relative_product(converse(R1), R2).

 + Return a restriction of a binary relation.

Returns the restriction of - the binary relation BinRel1 - to Set.

+ the binary relation BinRel1 + to Set.

 1> R1 = sofs:relation([{1,a},{2,b},{3,c}]),
 S = sofs:set([1,2,4]),
@@ -1397,13 +1512,14 @@ true
[{1,a},{2,b}] + Return a restriction of a set. -

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

+

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]),
@@ -1412,28 +1528,30 @@ true
[{2,b},{3,c}] + Create a set of atoms or any type of sets. -

Creates an unordered - set. set(L, T) is equivalent to +

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.

+ explicitly specified, [atom] is used as the set type.

 + Select a subset using a predicate.

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 + 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 + each element, otherwise Fun is applied to each element.

 1> R1 = sofs:relation([{a,1},{b,2}]),
@@ -1444,14 +1562,15 @@ true
[[{a,1},{b,2}]] + - Return the strict relation corresponding to + Return the strict relation corresponding to a given relation. -

Returns the strict +

Returns the strict relation corresponding to the binary - relation BinRel1.

+ relation BinRel1.

 1> R1 = sofs:relation([{1,1},{1,2},{2,1},{2,2}]),
 R2 = sofs:strict_relation(R1),
@@ -1459,13 +1578,14 @@ true
[{1,2},{2,1}] + Return a function with a given set as domain.

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 + 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}].
@@ -1482,24 +1602,24 @@ true
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:

+

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).
-

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 +

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 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:

+ 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),
@@ -1507,13 +1627,14 @@ images2(SetOfSets, BinRel) ->
    sofs:relation_to_family(R).
 + Return the symmetric difference of two sets. -

Returns the symmetric +

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

+ of Set1 and Set2.

 1> S1 = sofs:set([1,2,3]),
 S2 = sofs:set([2,3,4]),
@@ -1522,68 +1643,81 @@ images2(SetOfSets, BinRel) ->
 [1,4]
 + Return a partition of two sets. -

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.

+

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. + + + Return the elements of a set. -

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

+

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

 + - Return a list or a tuple of the elements of set. + Return a list or a tuple of the elements of a set. -

Returns the elements of the ordered set ASet +

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 + ASet as a sorted list of sets without duplicates.

 + Return the type of a set.

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

+ atomic, ordered, or unordered set.

 + Return the union of a set of sets.

Returns the union of the - set of sets SetOfSets.

+ set of sets SetOfSets.

 + Return the union of two sets.

Returns the union of - Set1 and Set2.

+ Set1 and Set2.

 + Return the union of a family. -

Returns the union of - the family Family. -

+

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),
@@ -1591,16 +1725,17 @@ images2(SetOfSets, BinRel) ->
 [0,1,2,3,4]
 + - Return the weak relation corresponding to + Return the weak relation corresponding to a given relation.

Returns a subset S of the weak relation W - corresponding to the binary relation BinRel1. + corresponding to the binary relation BinRel1. Let F be the field of - BinRel1. The + 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.

@@ -1614,11 +1749,11 @@ images2(SetOfSets, BinRel) ->
 
   

     See Also
-    
dict(3), 
-      digraph(3),
-      orddict(3), 
-      ordsets(3), 
-      sets(3)

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

   

 
 
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