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<A HREF="http://www.erlang.se"><IMG BORDER=0 ALT="[Erlang Systems]" SRC="min_head.gif"></A>
<H1>sofs</H1>
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<H3>MODULE</H3>
<UL>
sofs</UL>
<H3>MODULE SUMMARY</H3>
<UL>
Functions for Manipulating Sets of Sets</UL>
<H3>DESCRIPTION</H3>
<UL>
<P>The <CODE>sofs</CODE> module implements operations on finite sets and
relations represented as sets. Intuitively, a set is a
collection of elements; every element belongs to the set, and
the set contains every element.

<P>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&#160;=
{x&#160;in&#160;A&#160;: S(x)}. Sentences are expressed using
the logical operators &#34;for some&#34; (or &#34;there exists&#34;), &#34;for all&#34;,
&#34;and&#34;, &#34;or&#34;, &#34;not&#34;. 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&#160;= {x&#160;: S(x)}. 

<P>The <STRONG>unordered set</STRONG> containing the elements a, b and c
is denoted {a,&#160;b,&#160;c}. This notation is not to be
confused with tuples. The <STRONG>ordered pair</STRONG> of a and b, with
first <STRONG>coordinate</STRONG> a and second coordinate b, is denoted
(a,&#160;b). An ordered pair is an <STRONG>ordered set</STRONG> 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.

<P>The set that contains no elements is called the <STRONG>empty
set</STRONG>. If two sets A and B contain the same elements, then A
is <A NAME="equal"><!-- Empty --></A><STRONG>equal</STRONG> to B, denoted
A&#160;=&#160;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 <A NAME="subset"><!-- Empty --></A><STRONG>subset</STRONG> of A. The
<A NAME="union"><!-- Empty --></A><STRONG>union</STRONG> of two sets A and B is the
smallest set that contains all elements of A and all elements of
B. The <A NAME="intersection"><!-- Empty --></A><STRONG>intersection</STRONG> of two
sets A and B is the set that contains all elements of A that
belong to B. Two sets are <A NAME="disjoint"><!-- Empty --></A><STRONG>disjoint</STRONG> if their intersection is the
empty set. The <A NAME="difference"><!-- Empty --></A><STRONG>difference</STRONG> of
two sets A and B is the set that contains all elements of A that
do not belong to B. The <A NAME="symmetric_difference"><!-- Empty --></A><STRONG>symmetric difference</STRONG> of two
sets is the set that contains those element that belong to
either of the two sets, but not both. The <A NAME="union_n"><!-- Empty --></A><STRONG>union</STRONG> 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 <A NAME="intersection_n"><!-- Empty --></A><STRONG>intersection</STRONG> of a non-empty
collection of sets is the set that contains all elements that
belong to every set of the collection.

<P>The <A NAME="Cartesian_product"><!-- Empty --></A><STRONG>Cartesian product</STRONG>
of two sets X and Y, denoted X&#160;&#215;&#160;Y, is the set
{a&#160;: a&#160;= (x,&#160;y) for some x&#160;in&#160;X and for
some y&#160;in&#160;Y}. A <A NAME="relation"><!-- Empty --></A><STRONG>relation</STRONG> is a subset of
X&#160;&#215;&#160;Y. Let R be a relation. The fact that
(x,&#160;y) belongs to R is written as x&#160;R&#160;y. Since
relations are sets, the definitions of the last paragraph
(subset, union, and so on) apply to relations as well. The
<A NAME="domain"><!-- Empty --></A><STRONG>domain</STRONG> of R is the set {x&#160;:
x&#160;R&#160;y for some y&#160;in&#160;Y}. The <A NAME="range"><!-- Empty --></A><STRONG>range</STRONG> of R is the set {y&#160;:
x&#160;R&#160;y for some x&#160;in&#160;X}. The <A NAME="converse"><!-- Empty --></A><STRONG>converse</STRONG> of R is the set {a&#160;:
a&#160;= (y,&#160;x) for some (x,&#160;y)&#160;in&#160;R}. If A
is a subset of X, then the <A NAME="image"><!-- Empty --></A><STRONG>image</STRONG> of
A under R is the set {y&#160;: x&#160;R&#160;y for some
x&#160;in&#160;A}, and if B is a subset of Y, then the <A NAME="inverse_image"><!-- Empty --></A><STRONG>inverse image</STRONG> of B is the set
{x&#160;: x&#160;R&#160;y for some y&#160;in&#160;B}. If R is a
relation from X to Y and S is a relation from Y to Z, then the
<A NAME="relative_product"><!-- Empty --></A><STRONG>relative product</STRONG> of R and
S is the relation T from X to Z defined so that x&#160;T&#160;z
if and only if there exists an element y in Y such that
x&#160;R&#160;y and y&#160;S&#160;z. The <A NAME="restriction"><!-- Empty --></A><STRONG>restriction</STRONG> of R to A is the set S
defined so that x&#160;S&#160;y if and only if there exists an
element x in A such that x&#160;R&#160;y. If X&#160;=&#160;Y
then we call R a relation <STRONG>in</STRONG> X. The <A NAME="field"><!-- Empty --></A><STRONG>field</STRONG> 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&#160;S&#160;y if x&#160;R&#160;y and
not x&#160;=&#160;y, then S is the <A NAME="strict_relation"><!-- Empty --></A><STRONG>strict</STRONG> relation corresponding to
R, and vice versa, if S is a relation in X, and if R is defined
so that x&#160;R&#160;y if x&#160;S&#160;y or x&#160;=&#160;y,
then R is the <A NAME="weak_relation"><!-- Empty --></A><STRONG>weak</STRONG> relation
corresponding to S. A relation R in X is <STRONG>reflexive</STRONG> if
x&#160;R&#160;x for every element x of X; it is
<STRONG>symmetric</STRONG> if x&#160;R&#160;y implies that
y&#160;R&#160;x; and it is <STRONG>transitive</STRONG> if
x&#160;R&#160;y and y&#160;R&#160;z imply that x&#160;R&#160;z.

<P>A <A NAME="function"><!-- Empty --></A><STRONG>function</STRONG> F is a relation, a
subset of X&#160;&#215;&#160;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,&#160;y) in F. The latter condition can
be formulated as follows: if x&#160;F&#160;y and x&#160;F&#160;z
then y&#160;=&#160;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,&#160;y)&#160;in&#160;F or x&#160;F&#160;y, we write
F(x)&#160;=&#160;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 <A NAME="inverse"><!-- Empty --></A><STRONG>inverse</STRONG> of F. The relative product of two
functions F1 and F2 is called the <A NAME="composite"><!-- Empty --></A><STRONG>composite</STRONG> of F1 and F2 if the range of
F1 is a subset of the domain of F2. 

<P>Sometimes, when the range of a function is more important than
the function itself, the function is called a <STRONG>family</STRONG>.
The domain of a family is called the <STRONG>index set</STRONG>, and the
range is called the <STRONG>indexed set</STRONG>. If x is a family from
I to X, then x[i] denotes the value of the function at index i.
The notation &#34;a family in X&#34; is used for such a family. When the
indexed set is a set of subsets of a set X, then we call x a
<A NAME="family"><!-- Empty --></A><STRONG>family of subsets</STRONG> of X. If x is a
family of subsets of X, then the union of the range of x is
called the <STRONG>union of the family</STRONG> x. If x is non-empty
(the index set is non-empty), the <STRONG>intersection of the
family</STRONG> 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 &#34;family&#34;
will be used for such families of subsets.

<P>A <A NAME="partition"><!-- Empty --></A><STRONG>partition</STRONG> 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
<STRONG>equivalence relation</STRONG> if it is reflexive, symmetric and
transitive. If R is an equivalence relation in X, and x is an
element of X, the <A NAME="equivalence_class"><!-- Empty --></A><STRONG>equivalence
class</STRONG> of x with respect to R is the set of all those
elements y of X for which x&#160;R&#160;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 <A NAME="canonical_map"><!-- Empty --></A><STRONG>canonical map</STRONG> is the function that
maps every element of X onto its equivalence class.

<P>Relations as defined above (as sets of ordered pairs) will from
now on be referred to as <STRONG>binary relations</STRONG>. We call a
set of ordered sets (x[1],&#160;...,&#160;x[n]) an <STRONG>(n-ary)
relation</STRONG>, and say that the relation is a subset of the
<A NAME="Cartesian_product_tuple"><!-- Empty --></A> Cartesian product
X[1]&#160;&#215;&#160;...&#160;&#215;&#160;X[n] where x[i] is
an element of X[i], 1&#160;&#60;=&#160;i&#160;&#60;=&#160;n. The
<A NAME="projection"><!-- Empty --></A><STRONG>projection</STRONG> of an n-ary relation
R onto coordinate i is the set {x[i]&#160;:
(x[1],&#160;...,&#160;x[i],&#160;...,&#160;x[n]) in R for some
x[j]&#160;in&#160;X[j], 1&#160;&#60;=&#160;j&#160;&#60;=&#160;n
and not i&#160;=&#160;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 a an ordered set (R[1],&#160;...,&#160;R[n]) of binary
relations from X to Y[i] and S a binary relation from
(Y[1]&#160;&#215;&#160;...&#160;&#215;&#160;Y[n]) to Z. The
<A NAME="tuple_relative_product"><!-- Empty --></A><STRONG>relative product</STRONG> of
TR and S is the binary relation T from X to Z defined so that
x&#160;T&#160;z if and only if there exists an element y[i] in
Y[i] for each 1&#160;&#60;=&#160;i&#160;&#60;=&#160;n such that
x&#160;R[i]&#160;y[i] and
(y[1],&#160;...,&#160;y[n])&#160;S&#160;z. Now let TR be a an
ordered set (R[1],&#160;...,&#160;R[n]) of binary relations from
X[i] to Y[i] and S a subset of
X[1]&#160;&#215;&#160;...&#160;&#215;&#160;X[n]. The <A NAME="multiple_relative_product"><!-- Empty --></A><STRONG>multiple relative
product</STRONG> of TR and and S is defined to be the set {z&#160;:
z&#160;= ((x[1],&#160;...,&#160;x[n]), (y[1],...,y[n])) for some
(x[1],&#160;...,&#160;x[n])&#160;in&#160;S and for some
(x[i],&#160;y[i]) in R[i],
1&#160;&#60;=&#160;i&#160;&#60;=&#160;n}. The <A NAME="natural_join"><!-- Empty --></A><STRONG>natural join</STRONG> of an n-ary relation R
and an m-ary relation S on coordinate i and j is defined to be
the set {z&#160;: z&#160;= (x[1],&#160;...,&#160;x[n],&#160;
y[1],&#160;...,&#160;y[j-1],&#160;y[j+1],&#160;...,&#160;y[m])
for some (x[1],&#160;...,&#160;x[n])&#160;in&#160;R and for some
(y[1],&#160;...,&#160;y[m])&#160;in&#160;S such that
x[i]&#160;=&#160;y[j]}.

<P><A NAME="sets_definition"><!-- Empty --></A>The sets recognized by this module
will be represented by elements of the relation Sets, defined as
the smallest set such that:

<P><UL>
<LI>for every atom T except '_' and for every term X,
(T,&#160;X) belongs to Sets (<STRONG>atomic sets</STRONG>);

</LI><BR>
<LI>(['_'],&#160;[]) belongs to Sets (the <STRONG>untyped empty
set</STRONG>);

</LI><BR>
<LI>for every tuple T&#160;= {T[1],&#160;...,&#160;T[n]} and
for every tuple X&#160;= {X[1],&#160;...,&#160;X[n]}, if
(T[i],&#160;X[i]) belongs to Sets for every
1&#160;&#60;=&#160;i&#160;&#60;=&#160;n then (T,&#160;X) belongs
to Sets (<STRONG>ordered sets</STRONG>);

</LI><BR>
<LI>for every term T, if X is the empty list or a non-empty
sorted list [X[1],&#160;...,&#160;X[n]] without duplicates
such that (T,&#160;X[i]) belongs to Sets for every
1&#160;&#60;=&#160;i&#160;&#60;=&#160;n, then ([T],&#160;X)
belongs to Sets (<STRONG>typed unordered sets</STRONG>).
</LI><BR>
</UL>
<P>An <A NAME="external_set"><!-- Empty --></A><STRONG>external set</STRONG> is an
element in the range of Sets. A <A NAME="type"><!-- Empty --></A><STRONG>type</STRONG>
is an element in the domain of Sets. If S is an element
(T,&#160;X) of Sets, then T is a <A NAME="valid_type"><!-- Empty --></A><STRONG>valid type</STRONG> of X, T is the type of S,
and X is the external set of S. <A HREF="#from_term">from_term/2</A> creates a set from a
type and an Erlang term turned into an external set.

<P>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:

<P><UL>
<LI>Set(T,Term)&#160;= Term, where T is an atom;
</LI><BR>
<LI>Set({T[1],&#160;...,&#160;T[n]},&#160;{X[1],&#160;...,&#160;X[n]})
&#160;= (Set(T[1],&#160;X[1]),&#160;...,&#160;Set(T[n],&#160;X[n]));
</LI><BR>
<LI>Set([T],&#160;[X[1],&#160;...,&#160;X[n]])
&#160;= {Set(T,&#160;X[1]),&#160;...,&#160;Set(T,&#160;X[n])};
</LI><BR>
<LI>Set([T],&#160;[])&#160;= {}.
</LI><BR>
</UL>
<P>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 <CODE>union/2</CODE> 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).

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

<P>A few functions of this module (<CODE>drestriction/3</CODE>,
<CODE>family_projection/2</CODE>, <CODE>partition/2</CODE>,
<CODE>partition_family/2</CODE>, <CODE>projection/2</CODE>,
<CODE>restriction/3</CODE>, <CODE>substitution/2</CODE>) accept Erlang
functions 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 function, a tuple <CODE>{external, Fun}</CODE>, or an
integer. The two latter alternatives are optimizations; instead
of a set, the argument presented to the function is an external
set, which in the present implementation can be done more
efficiently. This optimization can however only be applied 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 <CODE>{external, fun(X)&#160;-&#62; element(I,&#160;X)}</CODE>,
but is to be preferred since it makes it possible to handle this
case even more efficiently. Examples of SetFuns:

<PRE>{sofs, union}
fun(S) -&#62; sofs:partition(1, S) end
{external, fun(A) -&#62; A end}
{external, fun({A,_,C}) -&#62; {C,A} end}
{external, fun({_,{_,C}}) -&#62; C end}
{external, fun({_,{_,{_,E}=C}}) -&#62; {E,{E,C}} end}
2</PRE>
<P>The order in which functions are applied to elements is not
specified, and may change in future versions of sofs.

<P>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: <CODE>from_external</CODE>,
<CODE>is_empty_set</CODE>, <CODE>is_set</CODE>, <CODE>is_sofs_set</CODE>,
<CODE>to_external</CODE>, <CODE>type</CODE>.

<P>The functions of this module exit the process with a
<CODE>badarg</CODE>, <CODE>bad_function</CODE>, or <CODE>type_mismatch</CODE>
message when given badly formed arguments or sets the types of
which are not compatible.

<P><STRONG>Types</STRONG> 
<PRE>anyset() = -&#160;an unordered, ordered or atomic set&#160;-
binary_relation() = -&#160;a binary relation&#160;-
bool() = true | false
external_set() = -&#160;an external set&#160;-
family() = -&#160;a family (of subsets)&#160;-
function() = -&#160;a function&#160;-
ordset() = -&#160;an ordered set&#160;-
relation() = -&#160;an n-ary relation&#160;-
set() = -&#160;an unordered set&#160;-
set_of_sets() = -&#160;an unordered set of set()&#160;-
set_fun() = integer() &#62;= 1
          | {external, fun(external_set()) -&#62; external_set()}
          | fun(anyset()) -&#62; anyset()
spec_fun() = {external, fun(external_set()) -&#62; bool()}
           | fun(anyset()) -&#62; bool()
type() = -&#160;a type&#160;-</PRE>
</UL>
<H3>EXPORTS</H3>
<P><A NAME="a_function%2"><STRONG><CODE>a_function(Tuples [, Type]) -&#62; Function</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>Function = function()</CODE></STRONG><BR>
<STRONG><CODE>Tuples = [tuple()]</CODE></STRONG><BR>
<STRONG><CODE>Type = type()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Creates a <A HREF="#function">function</A>.
<CODE>a_function(F, T)</CODE> is equivalent to <CODE>from_term(F,
T)</CODE>, if the result is a function. If no <A HREF="#type">type</A> is explicitly given,
<CODE>[{atom,&#160;atom}]</CODE> is used as type of the function.
</UL>
<P><A NAME="canonical_relation%1"><STRONG><CODE>canonical_relation(SetOfSets) -&#62; BinRel</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>BinRel = binary_relation()</CODE></STRONG><BR>
<STRONG><CODE>SetOfSets = set_of_sets()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Returns the binary relation containing the elements
         (E,&#160;Set) such that Set belongs to SetOfSets and E
         belongs to Set. If SetOfSets is a <A HREF="#partition">partition</A> of a set X and R is
         the equivalence relation in X induced by SetOfSets, then the
         returned relation is the <A HREF="#canonical_map">canonical map</A> from X onto
         the equivalence classes with respect to R.

        <PRE>1&#62; <STRONG>Ss = sofs:from_term([[a,b],[b,c]]),</STRONG>
<STRONG>CR = sofs:canonical_relation(Ss),</STRONG>
<STRONG>sofs:to_external(CR).</STRONG>
[{a,[a,b]},{b,[a,b]},{b,[b,c]},{c,[b,c]}]</PRE>
</UL>
<P><A NAME="composite%2"><STRONG><CODE>composite(Function1, Function2) -&#62; Function3</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>Function1 = Function2 = Function3 = function()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Returns the <A HREF="#composite">composite</A> of the functions
         Function1 and Function2.

        <PRE>1&#62; <STRONG>F1 = sofs:a_function([{a,1},{b,2},{c,2}]),</STRONG>
<STRONG>F2 = sofs:a_function([{1,x},{2,y},{3,z}]),</STRONG>
<STRONG>F = sofs:composite(F1, F2),</STRONG>
<STRONG>sofs:to_external(F).</STRONG>
[{a,x},{b,y},{c,y}]</PRE>
</UL>
<P><A NAME="constant_function%2"><STRONG><CODE>constant_function(Set, AnySet) -&#62; Function</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>AnySet = anyset()</CODE></STRONG><BR>
<STRONG><CODE>Function = function()</CODE></STRONG><BR>
<STRONG><CODE>Set = set()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Creates the <A HREF="#function">function</A>
         that maps each element of the set Set onto AnySet.

        <PRE>1&#62; <STRONG>S = sofs:set([a,b]),</STRONG>
<STRONG>E = sofs:from_term(1),</STRONG>
<STRONG>R = sofs:constant_function(S, E),</STRONG>
<STRONG>sofs:to_external(R).</STRONG>
[{a,1},{b,1}]</PRE>
</UL>
<P><A NAME="converse%1"><STRONG><CODE>converse(BinRel1) -&#62; BinRel2</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>BinRel1 = BinRel2 = binary_relation()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Returns the <A HREF="#converse">converse</A>
of the binary relation BinRel1.

        <PRE>1&#62; <STRONG>R1 = sofs:relation([{1,a},{2,b},{3,a}]),</STRONG>
<STRONG>R2 = sofs:converse(R1),</STRONG>
<STRONG>sofs:to_external(R2).</STRONG>
[{a,1},{a,3},{b,2}]</PRE>
</UL>
<P><A NAME="difference%2"><STRONG><CODE>difference(Set1, Set2) -&#62; Set3</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>Set1 = Set2 = Set3 = set()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Returns the <A HREF="#difference">difference
         </A> of the sets Set1 and Set2.
</UL>
<P><A NAME="digraph_to_family%2"><STRONG><CODE>digraph_to_family(Graph [, Type]) -&#62; Family</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>Graph = digraph() -&#160;see digraph(3)&#160;-</CODE></STRONG><BR>
<STRONG><CODE>Family = family()</CODE></STRONG><BR>
<STRONG><CODE>Type = type()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Creates a <A HREF="#family">family</A> from
the directed graph Graph. Each vertex a of Graph is
represented by a pair (a,&#160;{b[1],&#160;...,&#160;b[n]})
where the b[i]'s are the out-neighbours of a.
<CODE>digraph_to_family(G)</CODE> is equivalent to
<CODE>digraph_to_family(G, [{atom,&#160;[atom]}])</CODE>. It is
assumed that Type is a <A HREF="#valid_type">valid
type</A> of the external set of the family.

        <P>If G is a directed graph, it holds that the vertices and
edges of G are the same as the vertices and edges of
<CODE>family_to_digraph(digraph_to_family(G))</CODE>.
</UL>
<P><A NAME="domain%1"><STRONG><CODE>domain(BinRel) -&#62; Set</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>BinRel = binary_relation()</CODE></STRONG><BR>
<STRONG><CODE>Set = set()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Returns the <A HREF="#domain">domain
         </A> of the binary relation BinRel.

        <PRE>1&#62; <STRONG>R = sofs:relation([{1,a},{1,b},{2,b},{2,c}]),</STRONG>
<STRONG>S = sofs:domain(R),</STRONG>
<STRONG>sofs:to_external(S).</STRONG>
[1,2]</PRE>
</UL>
<P><A NAME="drestriction%2"><STRONG><CODE>drestriction(BinRel1, Set) -&#62; BinRel2</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>BinRel1 = BinRel2 = binary_relation()</CODE></STRONG><BR>
<STRONG><CODE>Set = set()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Returns the difference between the binary relation BinRel1
         and the <A HREF="#restriction">restriction</A>
         of BinRel1 to Set; <CODE>drestriction(R, S)</CODE> is equivalent
         to <CODE>difference(R, restriction(R, S))</CODE>.

        <PRE>1&#62; <STRONG>R1 = sofs:relation([{1,a},{2,b},{3,c}]),</STRONG>
<STRONG>S = sofs:set([2,4,6]),</STRONG>
<STRONG>R2 = sofs:drestriction(R1, S),</STRONG>
<STRONG>sofs:to_external(R2).</STRONG>
[{1,a},{3,c}]</PRE>
</UL>
<P><A NAME="drestriction%3"><STRONG><CODE>drestriction(SetFun, Set1, Set2) -&#62; Set3</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>SetFun = set_fun()</CODE></STRONG><BR>
<STRONG><CODE>Set1 = Set2 = Set3 = set()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Returns a subset of Set1 containing those elements that do
not yield an element in Set2 as the result of applying
SetFun; <CODE>drestriction(F, S1, S2)</CODE> is equivalent to
<CODE>difference(S1, restriction(F, S1, S2))</CODE>.

        <PRE>1&#62; <STRONG>SetFun = {external, fun({_A,B,C}) -&#62; {B,C} end},</STRONG>
<STRONG>R1 = sofs:relation([{a,aa,1},{b,bb,2},{c,cc,3}]),</STRONG>
<STRONG>R2 = sofs:relation([{bb,2},{cc,3},{dd,4}]),</STRONG>
<STRONG>R3 = sofs:drestriction(SetFun, R1, R2),</STRONG>
<STRONG>sofs:to_external(R3).</STRONG>
[{a,aa,1}]</PRE>
</UL>
<P><A NAME="empty_set%0"><STRONG><CODE>empty_set() -&#62; Set</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>Set = set()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Returns the <A HREF="#sets_definition">untyped
empty set</A>. <CODE>empty_set()</CODE> is equivalent to
<CODE>from_term([], ['_'])</CODE>.
</UL>
<P><A NAME="family%2"><STRONG><CODE>family(Tuples [, Type]) -&#62; Family</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>Family = family()</CODE></STRONG><BR>
<STRONG><CODE>Tuples = [tuple()]</CODE></STRONG><BR>
<STRONG><CODE>Type = type()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Creates a <A HREF="#family">family of
subsets</A>. <CODE>family(F, T)</CODE> is equivalent to
<CODE>from_term(F, T)</CODE>, if the result is a family. If no
<A HREF="#type">type</A> is explicitly given,
<CODE>[{atom,&#160;[atom]}]</CODE> is used as type of the family.
</UL>
<P><A NAME="family_difference%2"><STRONG><CODE>family_difference(Family1, Family2) -&#62; Family3</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>Family1 = Family2 = Family3 = family()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>If Family1 and Family2 are <A HREF="#family">families</A>, then Family3 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.

        <PRE>1&#62; <STRONG>F1 = sofs:family([{a,[1,2]},{b,[3,4]}]),</STRONG>
<STRONG>F2 = sofs:family([{b,[4,5]},{c,[6,7]}]),</STRONG>
<STRONG>F3 = sofs:family_difference(F1, F2),</STRONG>
<STRONG>sofs:to_external(F3).</STRONG>
[{a,[1,2]},{b,[3]}]</PRE>
</UL>
<P><A NAME="family_domain%1"><STRONG><CODE>family_domain(Family1) -&#62; Family2</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>Family1 = Family2 = family()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>If Family1 is a <A HREF="#family">family</A>
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 <A HREF="#domain">domain</A> of Family1[i].

        <PRE>1&#62; <STRONG>FR = sofs:from_term([{a,[{1,a},{2,b},{3,c}]},{b,[]},{c,[{4,d},{5,e}]}]),</STRONG>
<STRONG>F = sofs:family_domain(FR),</STRONG>
<STRONG>sofs:to_external(F).</STRONG>
[{a,[1,2,3]},{b,[]},{c,[4,5]}]</PRE>
</UL>
<P><A NAME="family_field%1"><STRONG><CODE>family_field(Family1) -&#62; Family2</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>Family1 = Family2 = family()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>If Family1 is a <A HREF="#family">family</A>
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 <A HREF="#field">field</A> of Family1[i];
<CODE>family_field(Family1)</CODE> is equivalent to
<CODE>family_union(family_domain(Family1),
family_range(Family1))</CODE>.

        <PRE>1&#62; <STRONG>FR = sofs:from_term([{a,[{1,a},{2,b},{3,c}]},{b,[]},{c,[{4,d},{5,e}]}]),</STRONG>
<STRONG>F = sofs:family_field(FR),</STRONG>
<STRONG>sofs:to_external(F).</STRONG>
[{a,[1,2,3,a,b,c]},{b,[]},{c,[4,5,d,e]}]</PRE>
</UL>
<P><A NAME="family_intersection%1"><STRONG><CODE>family_intersection(Family1) -&#62; Family2</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>Family1 = Family2 = family()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>If Family1 is a <A HREF="#family">family</A>
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 <A HREF="#intersection_n">intersection</A> of
Family1[i].

        <P>If Family1[i] is an empty set for some i, then the process
         exits with a <CODE>badarg</CODE> message.

        <PRE>1&#62; <STRONG>F1 = sofs:from_term([{a,[[1,2,3],[2,3,4]]},{b,[[x,y,z],[x,y]]}]),</STRONG>
<STRONG>F2 = sofs:family_intersection(F1),</STRONG>
<STRONG>sofs:to_external(F2).</STRONG>
[{a,[2,3]},{b,[x,y]}]</PRE>
</UL>
<P><A NAME="family_intersection%2"><STRONG><CODE>family_intersection(Family1, Family2) -&#62; Family3</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>Family1 = Family2 = Family3 = family()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>If Family1 and Family2 are <A HREF="#family">families</A>, 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].

        <PRE>1&#62; <STRONG>F1 = sofs:family([{a,[1,2]},{b,[3,4]},{c,[5,6]}]),</STRONG>
<STRONG>F2 = sofs:family([{b,[4,5]},{c,[7,8]},{d,[9,10]}]),</STRONG>
<STRONG>F3 = sofs:family_intersection(F1, F2),</STRONG>
<STRONG>sofs:to_external(F3).</STRONG>
[{b,[4]},{c,[]}]</PRE>
</UL>
<P><A NAME="family_projection%2"><STRONG><CODE>family_projection(SetFun, Family1) -&#62; Family2</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>SetFun = set_fun()</CODE></STRONG><BR>
<STRONG><CODE>Family1 = Family2 = family()</CODE></STRONG><BR>
<STRONG><CODE>Set = set()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>If Family1 is a <A HREF="#family">family</A>
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.

        <PRE>1&#62; <STRONG>F1 = sofs:from_term([{a,[[1,2],[2,3]]},{b,[[]]}]),</STRONG>
<STRONG>F2 = sofs:family_projection({sofs, union}, F1),</STRONG>
<STRONG>sofs:to_external(F2).</STRONG>
[{a,[1,2,3]},{b,[]}]</PRE>
</UL>
<P><A NAME="family_range%1"><STRONG><CODE>family_range(Family1) -&#62; Family2</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>Family1 = Family2 = family()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>If Family1 is a <A HREF="#family">family</A>
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 <A HREF="#range">range</A> of Family1[i].

        <PRE>1&#62; <STRONG>FR = sofs:from_term([{a,[{1,a},{2,b},{3,c}]},{b,[]},{c,[{4,d},{5,e}]}]),</STRONG>
<STRONG>F = sofs:family_range(FR),</STRONG>
<STRONG>sofs:to_external(F).</STRONG>
[{a,[a,b,c]},{b,[]},{c,[d,e]}]</PRE>
</UL>
<P><A NAME="family_specification%2"><STRONG><CODE>family_specification(Fun, Family1) -&#62; Family2</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>Fun = spec_fun()</CODE></STRONG><BR>
<STRONG><CODE>Family1 = Family2 = family()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>If Family1 is a <A HREF="#family">family</A>,
then Family2 is the <A HREF="#restriction"> restriction</A> of Family1 to those elements i of the
index set for which Fun applied to Family1[i] returns
<CODE>true</CODE>. If Fun is a tuple <CODE>{external, Fun2}</CODE>, Fun2
is applied to the <A HREF="#external_set">external
set</A> of Family1[i], otherwise Fun is applied to
Family1[i].

        <PRE>1&#62; <STRONG>F1 = sofs:family([{a,[1,2,3]},{b,[1,2]},{c,[1]}]),</STRONG>
<STRONG>SpecFun = fun(S) -&#62; sofs:no_elements(S) =:= 2 end,</STRONG>
<STRONG>F2 = sofs:family_specification(SpecFun, F1),</STRONG>
<STRONG>sofs:to_external(F2).</STRONG>
[{b,[1,2]}]</PRE>
</UL>
<P><A NAME="family_to_digraph%2"><STRONG><CODE>family_to_digraph(Family [, GraphType]) -&#62; Graph</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>Graph = digraph()</CODE></STRONG><BR>
<STRONG><CODE>Family = family()</CODE></STRONG><BR>
<STRONG><CODE>GraphType = -&#160;see digraph(3)&#160;-</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Creates a directed graph from the <A HREF="#family">family</A> Family. For each pair
(a,&#160;{b[1],&#160;...,&#160;b[n]}) of Family, the vertex
a as well the edges (a,&#160;b[i]) for
1&#160;&#60;=&#160;i&#160;&#60;=&#160;n are added to a newly
created directed graph.

        <P>If no graph type is given, <CODE>digraph:new/1</CODE> is used for
         creating the directed graph, otherwise the GraphType
         argument is passed on as second argument to
         <CODE>digraph:new/2</CODE>.

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

        <P>Creating a cycle in an acyclic graph exits the process with
         a <CODE>cyclic</CODE> message.
</UL>
<P><A NAME="family_to_relation%1"><STRONG><CODE>family_to_relation(Family) -&#62; BinRel</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>Family = family()</CODE></STRONG><BR>
<STRONG><CODE>BinRel = binary_relation()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>If Family is a <A HREF="#family">family</A>,
         then BinRel is the binary relation containing all pairs
         (i,&#160;x) such that i belongs to the index set of Family
         and x belongs to Family[i].

        <PRE>1&#62; <STRONG>F = sofs:family([{a,[]}, {b,[1]}, {c,[2,3]}]),</STRONG>
<STRONG>R = sofs:family_to_relation(F),</STRONG>
<STRONG>sofs:to_external(R).</STRONG>
[{b,1},{c,2},{c,3}]</PRE>
</UL>
<P><A NAME="family_union%1"><STRONG><CODE>family_union(Family1) -&#62; Family2</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>Family1 = Family2 = family()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>If Family1 is a <A HREF="#family">family</A>
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 <A HREF="#union_n">union</A> of Family1[i];
<CODE>family_union(F)</CODE> is equivalent to
<CODE>family_projection({sofs,union}, F)</CODE>.

        <PRE>1&#62; <STRONG>F1 = sofs:from_term([{a,[[1,2],[2,3]]},{b,[[]]}]),</STRONG>
<STRONG>F2 = sofs:family_union(F1),</STRONG>
<STRONG>sofs:to_external(F2).</STRONG>
[{a,[1,2,3]},{b,[]}]</PRE>
</UL>
<P><A NAME="family_union%2"><STRONG><CODE>family_union(Family1, Family2) -&#62; Family3</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>Family1 = Family2 = Family3 = family()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>If Family1 and Family2 are <A HREF="#family">families</A>, 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.

        <PRE>1&#62; <STRONG>F1 = sofs:family([{a,[1,2]},{b,[3,4]},{c,[5,6]}]),</STRONG>
<STRONG>F2 = sofs:family([{b,[4,5]},{c,[7,8]},{d,[9,10]}]),</STRONG>
<STRONG>F3 = sofs:family_union(F1, F2),</STRONG>
<STRONG>sofs:to_external(F3).</STRONG>
[{a,[1,2]},{b,[3,4,5]},{c,[5,6,7,8]},{d,[9,10]}]</PRE>
</UL>
<P><A NAME="field%1"><STRONG><CODE>field(BinRel) -&#62; Set</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>BinRel = binary_relation()</CODE></STRONG><BR>
<STRONG><CODE>Set = set()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Returns the <A HREF="#field">field</A> of the
         binary relation BinRel; <CODE>field(R)</CODE> is equivalent to
         <CODE>union(domain(R), range(R))</CODE>.

        <PRE>1&#62; <STRONG>R = sofs:relation([{1,a},{1,b},{2,b},{2,c}]),</STRONG>
<STRONG>S = sofs:field(R),</STRONG>
<STRONG>sofs:to_external(S).</STRONG>
[1,2,a,b,c]</PRE>
</UL>
<P><A NAME="from_external%2"><STRONG><CODE>from_external(ExternalSet, Type) -&#62; AnySet</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>ExternalSet = external_set()</CODE></STRONG><BR>
<STRONG><CODE>AnySet = anyset()</CODE></STRONG><BR>
<STRONG><CODE>Type = type()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Creates a set from the <A HREF="#external_set">external set</A> ExternalSet
         and the <A HREF="#type">type</A> Type. It is
         assumed that Type is a <A HREF="#valid_type">valid
         type</A> of ExternalSet.
</UL>
<P><A NAME="from_sets%1"><STRONG><CODE>from_sets(ListOfSets) -&#62; Set</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>Set = set()</CODE></STRONG><BR>
<STRONG><CODE>ListOfSets = [anyset()]</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Returns the <A HREF="#sets_definition">unordered
         set</A> containing the sets of the list ListOfSets.

        <PRE>1&#62; <STRONG>S1 = sofs:relation([{a,1},{b,2}]),</STRONG>
<STRONG>S2 = sofs:relation([{x,3},{y,4}]),</STRONG>
<STRONG>S = sofs:from_sets([S1,S2]),</STRONG>
<STRONG>sofs:to_external(S).</STRONG>
[[{a,1},{b,2}],[{x,3},{y,4}]]</PRE>
</UL>
<P><A NAME="from_sets%1"><STRONG><CODE>from_sets(TupleOfSets) -&#62; Ordset</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>Ordset = ordset()</CODE></STRONG><BR>
<STRONG><CODE>TupleOfSets = tuple-of(anyset())</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Returns the <A HREF="#sets_definition">ordered
         set</A> containing the sets of the non-empty tuple
         TupleOfSets.
</UL>
<P><A NAME="from_term%2"><STRONG><CODE>from_term(Term [, Type]) -&#62; AnySet</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>AnySet = anyset()</CODE></STRONG><BR>
<STRONG><CODE>Term = term()</CODE></STRONG><BR>
<STRONG><CODE>Type = type()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P><A NAME="from_term"><!-- Empty --></A>Creates an element of <A HREF="#sets_definition">Sets</A> by traversing the
term Term, sorting lists, removing duplicates and deriving
or verifying a <A HREF="#valid_type">valid
type</A> for the so obtained external set.
<CODE>from_term(T)</CODE> is equivalent to <CODE>from_term(T,
['_'])</CODE>. An explicitly given <A HREF="#type">type</A> Type can be used to limit the
depth of the traversal; an atomic type stops the traversal,
as demonstrated by this example where &#34;foo&#34; and {&#34;foo&#34;} are
left unmodified:

        <PRE>1&#62; <STRONG>S = sofs:from_term([{{&#34;foo&#34;},[1,1]},{&#34;foo&#34;,[2,2]}], [{atom,[atom]}]),</STRONG>
<STRONG>sofs:to_external(S).</STRONG>
[{{&#34;foo&#34;},[1]},{&#34;foo&#34;,[2]}]</PRE>
<P><CODE>from_term</CODE> can be used for creating atomic or ordered
         sets. The only purpose of such a set is that of later
         building unordered sets since all functions in this module
         that <STRONG>do</STRONG> anything operate on unordered sets.
         Creating unordered sets from a collection of ordered sets
         may be the way to go if the ordered sets are big and one
         does not want to waste heap by rebuilding the elements of
         the unordered set. An example showing that a set can be
         built &#34;layer by layer&#34;:
<PRE>1&#62; <STRONG>A = sofs:from_term(a),</STRONG>
<STRONG>S = sofs:set([1,2,3]),</STRONG>
<STRONG>P1 = sofs:from_sets({A,S}),</STRONG>
<STRONG>P2 = sofs:from_term({b,[6,5,4]}),</STRONG>
<STRONG>Ss = sofs:from_sets([P1,P2]),</STRONG>
<STRONG>sofs:to_external(Ss).</STRONG>
[{a,[1,2,3]},{b,[4,5,6]}]</PRE>
<P>Other functions that create sets are <CODE>from_external/2</CODE>
and <CODE>from_sets/1</CODE>. Special cases of <CODE>from_term/2</CODE>
are <CODE>a_function/1,2</CODE>, <CODE>empty_set/0</CODE>,
<CODE>family/1,2</CODE>, <CODE>relation/1,2</CODE>, and <CODE>set/1,2</CODE>.
</UL>
<P><A NAME="image%2"><STRONG><CODE>image(BinRel, Set1) -&#62; Set2</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>BinRel = binary_relation()</CODE></STRONG><BR>
<STRONG><CODE>Set1 = Set2 = set()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Returns the <A HREF="#image">image</A> of the
         set Set1 under the binary relation BinRel.
<PRE>1&#62; <STRONG>R = sofs:relation([{1,a},{2,b},{2,c},{3,d}]),</STRONG>
<STRONG>S1 = sofs:set([1,2]),</STRONG>
<STRONG>S2 = sofs:image(R, S1),</STRONG>
<STRONG>sofs:to_external(S2).</STRONG>
[a,b,c]</PRE>
</UL>
<P><A NAME="intersection%1"><STRONG><CODE>intersection(SetOfSets) -&#62; Set</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>Set = set()</CODE></STRONG><BR>
<STRONG><CODE>SetOfSets = set_of_sets()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Returns the <A HREF="#intersection_n">intersection</A> of the set
of sets SetOfSets.

        <P>Intersecting an empty set of sets exits the process with a
         <CODE>badarg</CODE> message.
</UL>
<P><A NAME="intersection%2"><STRONG><CODE>intersection(Set1, Set2) -&#62; Set3</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>Set1 = Set2 = Set3 = set()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Returns the <A HREF="#intersection">  intersection</A> of Set1 and Set2.
</UL>
<P><A NAME="intersection_of_family%1"><STRONG><CODE>intersection_of_family(Family) -&#62; Set</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>Family = family()</CODE></STRONG><BR>
<STRONG><CODE>Set = set()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Returns the intersection of the <A HREF="#family">family</A> Family.

        <P>Intersecting an empty family exits the process with a
         <CODE>badarg</CODE> message.

        <PRE>1&#62; <STRONG>F = sofs:family([{a,[0,2,4]},{b,[0,1,2]},{c,[2,3]}]),</STRONG>
<STRONG>S = sofs:intersection_of_family(F),</STRONG>
<STRONG>sofs:to_external(S).</STRONG>
[2]</PRE>
</UL>
<P><A NAME="inverse%1"><STRONG><CODE>inverse(Function1) -&#62; Function2</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>Function1 = Function2 = function()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Returns the <A HREF="#inverse">inverse</A>
of the function Function1. 

        <PRE>1&#62; <STRONG>R1 = sofs:relation([{1,a},{2,b},{3,c}]),</STRONG>
<STRONG>R2 = sofs:inverse(R1),</STRONG>
<STRONG>sofs:to_external(R2).</STRONG>
[{a,1},{b,2},{c,3}]</PRE>
</UL>
<P><A NAME="inverse_image%2"><STRONG><CODE>inverse_image(BinRel, Set1) -&#62; Set2</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>BinRel = binary_relation()</CODE></STRONG><BR>
<STRONG><CODE>Set1 = Set2 = set()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Returns the <A HREF="#inverse_image">inverse
         image</A> of Set1 under the binary relation BinRel.
<PRE>1&#62; <STRONG>R = sofs:relation([{1,a},{2,b},{2,c},{3,d}]),</STRONG>
<STRONG>S1 = sofs:set([c,d,e]),</STRONG>
<STRONG>S2 = sofs:inverse_image(R, S1),</STRONG>
<STRONG>sofs:to_external(S2).</STRONG>
[2,3]</PRE>
</UL>
<P><A NAME="is_a_function%1"><STRONG><CODE>is_a_function(BinRel) -&#62; Bool</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>Bool = bool()</CODE></STRONG><BR>
<STRONG><CODE>BinRel = binary_relation()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Returns <CODE>true</CODE> if the binary relation BinRel is a
         <A HREF="#function">function</A> or the
         untyped empty set, <CODE>false</CODE> otherwise.
</UL>
<P><A NAME="is_disjoint%2"><STRONG><CODE>is_disjoint(Set1, Set2) -&#62; Bool</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>Bool = bool()</CODE></STRONG><BR>
<STRONG><CODE>Set1 = Set2 = set()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Returns <CODE>true</CODE> if Set1 and Set2 are <A HREF="#disjoint">disjoint</A>, <CODE>false</CODE>
         otherwise.
</UL>
<P><A NAME="is_empty_set%1"><STRONG><CODE>is_empty_set(AnySet) -&#62; Bool</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>AnySet = anyset()</CODE></STRONG><BR>
<STRONG><CODE>Bool = bool()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Returns <CODE>true</CODE> if Set is an empty unordered set,
         <CODE>false</CODE> otherwise.
</UL>
<P><A NAME="is_equal%2"><STRONG><CODE>is_equal(AnySet1, AnySet2) -&#62; Bool</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>AnySet1 = AnySet2 = anyset()</CODE></STRONG><BR>
<STRONG><CODE>Bool = bool()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Returns <CODE>true</CODE> if the AnySet1 and AnySet2 are <A HREF="#equal">equal</A>, <CODE>false</CODE> otherwise.
</UL>
<P><A NAME="is_set%1"><STRONG><CODE>is_set(AnySet) -&#62; Bool</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>AnySet = anyset()</CODE></STRONG><BR>
<STRONG><CODE>Bool = bool()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Returns <CODE>true</CODE> if AnySet is an <A HREF="#sets_definition">unordered set</A>, and
         <CODE>false</CODE> if AnySet is an ordered set or an atomic set.
</UL>
<P><A NAME="is_sofs_set%1"><STRONG><CODE>is_sofs_set(Term) -&#62; Bool</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>Bool = bool()</CODE></STRONG><BR>
<STRONG><CODE>Term = term()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Returns <CODE>true</CODE> if Term is an <A HREF="#sets_definition">unordered set</A>, an
         ordered set or an atomic set, <CODE>false</CODE> otherwise.
</UL>
<P><A NAME="is_subset%2"><STRONG><CODE>is_subset(Set1, Set2) -&#62; Bool</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>Bool = bool()</CODE></STRONG><BR>
<STRONG><CODE>Set1 = Set2 = set()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Returns <CODE>true</CODE> if Set1 is a <A HREF="#subset">subset</A> of Set2, <CODE>false</CODE>
otherwise.
</UL>
<P><A NAME="is_type%1"><STRONG><CODE>is_type(Term) -&#62; Bool</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>Bool = bool()</CODE></STRONG><BR>
<STRONG><CODE>Term = term()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Returns <CODE>true</CODE> if the term Term is a <A HREF="#type">type</A>.
</UL>
<P><A NAME="join%4"><STRONG><CODE>join(Relation1, I, Relation2, J) -&#62; Relation3</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>Relation1 = Relation2 = Relation3 = relation()</CODE></STRONG><BR>
<STRONG><CODE>I = J = integer() &#62; 0</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Returns the <A HREF="#natural_join">natural
         join</A> of the relations Relation1 and Relation2 on
         coordinates I and J.

        <PRE>1&#62; <STRONG>R1 = sofs:relation([{a,x,1},{b,y,2}]),</STRONG>
<STRONG>R2 = sofs:relation([{1,f,g},{1,h,i},{2,3,4}]),</STRONG>
<STRONG>J = sofs:join(R1, 3, R2, 1),</STRONG>
<STRONG>sofs:to_external(J).</STRONG>
[{a,x,1,f,g},{a,x,1,h,i},{b,y,2,3,4}]</PRE>
</UL>
<P><A NAME="multiple_relative_product%2"><STRONG><CODE>multiple_relative_product(TupleOfBinRels, BinRel1) -&#62; 
BinRel2</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>TupleOfBinRels = tuple-of(BinRel)</CODE></STRONG><BR>
<STRONG><CODE>BinRel = BinRel1 = BinRel2 = binary_relation()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>If TupleOfBinRels is a non-empty tuple
         {R[1],&#160;...,&#160;R[n]} of binary relations and BinRel1
         is a binary relation, then BinRel2 is the <A HREF="#multiple_relative_product">multiple relative
         product</A> of the ordered set
         (R[i],&#160;...,&#160;R[n]) and BinRel1.

        <PRE>1&#62; <STRONG>Ri = sofs:relation([{a,1},{b,2},{c,3}]),</STRONG>
<STRONG>R = sofs:relation([{a,b},{b,c},{c,a}]),</STRONG>
<STRONG>MP = sofs:multiple_relative_product({Ri, Ri}, R),</STRONG>
<STRONG>sofs:to_external(sofs:range(MP)).</STRONG>
[{1,2},{2,3},{3,1}]</PRE>
</UL>
<P><A NAME="no_elements%1"><STRONG><CODE>no_elements(ASet) -&#62; NoElements</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>ASet = set() | ordset()</CODE></STRONG><BR>
<STRONG><CODE>NoElements = integer() &#62;= 0 </CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Returns the number of elements of the ordered or unordered
         set ASet.
</UL>
<P><A NAME="partition%1"><STRONG><CODE>partition(SetOfSets) -&#62; Partition</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>SetOfSets = set_of_sets()</CODE></STRONG><BR>
<STRONG><CODE>Partition = set()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Returns the <A HREF="#partition">partition</A> of the union of the
         set of sets SetOfSets such that two elements are considered
         equal if they are members of the same elements of SetOfSets.

        <PRE>1&#62; <STRONG>Sets1 = sofs:from_term([[a,b,c],[d,e,f],[g,h,i]]),</STRONG>
<STRONG>Sets2 = sofs:from_term([[b,c,d],[e,f,g],[h,i,j]]),</STRONG>
<STRONG>P = sofs:partition(sofs:union(Sets1, Sets2)),</STRONG>
<STRONG>sofs:to_external(P).</STRONG>
[[a],[b,c],[d],[e,f],[g],[h,i],[j]]</PRE>
</UL>
<P><A NAME="partition%2"><STRONG><CODE>partition(SetFun, Set) -&#62; Partition</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>SetFun = set_fun()</CODE></STRONG><BR>
<STRONG><CODE>Partition = set()</CODE></STRONG><BR>
<STRONG><CODE>Set = set()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Returns the <A HREF="#partition">partition</A> of Set such that two
         elements are considered equal if the results of applying
         SetFun are equal.

        <PRE>1&#62; <STRONG>Ss = sofs:from_term([[a],[b],[c,d],[e,f]]),</STRONG>
<STRONG>SetFun = fun(S) -&#62; sofs:from_term(sofs:no_elements(S)) end,</STRONG>
<STRONG>P = sofs:partition(SetFun, Ss),</STRONG>
<STRONG>sofs:to_external(P).</STRONG>
[[[a],[b]],[[c,d],[e,f]]]</PRE>
</UL>
<P><A NAME="partition_family%2"><STRONG><CODE>partition_family(SetFun, Set) -&#62; Family</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>Family = family()</CODE></STRONG><BR>
<STRONG><CODE>SetFun = set_fun()</CODE></STRONG><BR>
<STRONG><CODE>Set = set()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Returns the <A HREF="#family">family</A>
         Family where the indexed set is a <A HREF="#partition">partition</A> 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 <A HREF="#equivalence_class">equivalence class</A>.

        <PRE>1&#62; <STRONG>S = sofs:relation([{a,a,a,a},{a,a,b,b},{a,b,b,b}]),</STRONG>
<STRONG>SetFun = {external, fun({A,_,C,_}) -&#62; {A,C} end},</STRONG>
<STRONG>F = sofs:partition_family(SetFun, S),</STRONG>
<STRONG>sofs:to_external(F).</STRONG>
[{{a,a},[{a,a,a,a}]},{{a,b},[{a,a,b,b},{a,b,b,b}]}]</PRE>
</UL>
<P><A NAME="product%1"><STRONG><CODE>product(TupleOfSets) -&#62; Relation</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>Relation = relation()</CODE></STRONG><BR>
<STRONG><CODE>TupleOfSets = tuple-of(set())</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Returns the <A HREF="#Cartesian_product_tuple">       Cartesian product</A> of the non-empty tuple of sets
         TupleOfSets. If (x[1],&#160;...,&#160;x[n]) is an element of
         the n-ary relation Relation, then x[i] is drawn from element
         i of TupleOfSets.

        <PRE>1&#62; <STRONG>S1 = sofs:set([a,b]),</STRONG>
<STRONG>S2 = sofs:set([1,2]),</STRONG>
<STRONG>S3 = sofs:set([x,y]),</STRONG>
<STRONG>P3 = sofs:product({S1,S2,S3}),</STRONG>
<STRONG>sofs:to_external(P3).</STRONG>
[{a,1,x},{a,1,y},{a,2,x},{a,2,y},{b,1,x},{b,1,y},{b,2,x},{b,2,y}]</PRE>
</UL>
<P><A NAME="product%2"><STRONG><CODE>product(Set1, Set2) -&#62; BinRel</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>BinRel = binary_relation()</CODE></STRONG><BR>
<STRONG><CODE>Set1 = Set2 = set()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Returns the <A HREF="#Cartesian_product">Cartesian
product</A> of Set1 and Set2. <CODE>product(S1, S2)</CODE> is
         equivalent to <CODE>product({S1, S2})</CODE>.

        <PRE>1&#62; <STRONG>S1 = sofs:set([1,2]),</STRONG>
<STRONG>S2 = sofs:set([a,b]),</STRONG>
<STRONG>R = sofs:product(S1, S2),</STRONG>
<STRONG>sofs:to_external(R).</STRONG>
[{1,a},{1,b},{2,a},{2,b}]</PRE>
</UL>
<P><A NAME="projection%2"><STRONG><CODE>projection(SetFun, Set1) -&#62; Set2</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>SetFun = set_fun()</CODE></STRONG><BR>
<STRONG><CODE>Set1 = Set2 = set()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Returns the set created by substituting each element of
Set1 by the result of applying SetFun to the element.

<P>If SetFun is a number i&#160;&#62;=&#160;1 and Set1 is a
relation, then the returned set is the <A HREF="#projection">projection</A> of Set1 onto
coordinate i.

        <PRE>1&#62; <STRONG>S1 = sofs:from_term([{1,a},{2,b},{3,a}]),</STRONG>
<STRONG>S2 = sofs:projection(2, S1),</STRONG>
<STRONG>sofs:to_external(S2).</STRONG>
[a,b]</PRE>
</UL>
<P><A NAME="range%1"><STRONG><CODE>range(BinRel) -&#62; Set</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>BinRel = binary_relation()</CODE></STRONG><BR>
<STRONG><CODE>Set = set()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Returns the <A HREF="#range">range</A> of the
         binary relation BinRel.

        <PRE>1&#62; <STRONG>R = sofs:relation([{1,a},{1,b},{2,b},{2,c}]),</STRONG>
<STRONG>S = sofs:range(R),</STRONG>
<STRONG>sofs:to_external(S).</STRONG>
[a,b,c]</PRE>
</UL>
<P><A NAME="relation%2"><STRONG><CODE>relation(Tuples [, Type]) -&#62; Relation</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>N = integer()</CODE></STRONG><BR>
<STRONG><CODE>Type = N | type()</CODE></STRONG><BR>
<STRONG><CODE>Relation = relation()</CODE></STRONG><BR>
<STRONG><CODE>Tuples = [tuple()]</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Creates a <A HREF="#relation">relation</A>.
<CODE>relation(R, T)</CODE> is equivalent to <CODE>from_term(R,
T)</CODE>, if T is a <A HREF="#type">type</A> and
the result is a relation. If Type is an integer N, then
<CODE>[{atom,&#160;...,&#160;atom}])</CODE>, where the size of the
tuple is N, is used as type of the relation. If no type is
explicitly given, the size of the first tuple of Tuples is
used if there is such a tuple. <CODE>relation([])</CODE> is
equivalent to <CODE>relation([], 2)</CODE>.
</UL>
<P><A NAME="relation_to_family%1"><STRONG><CODE>relation_to_family(BinRel) -&#62; Family</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>Family = family()</CODE></STRONG><BR>
<STRONG><CODE>BinRel = binary_relation()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Returns the <A HREF="#family">family</A>
Family such that the index set is equal to the <A HREF="#domain">domain</A> of the binary relation
BinRel, and Family[i] is the <A HREF="#image">image</A> of the set of i under
BinRel.

        <PRE>1&#62; <STRONG>R = sofs:relation([{b,1},{c,2},{c,3}]),</STRONG>
<STRONG>F = sofs:relation_to_family(R),</STRONG>
<STRONG>sofs:to_external(F).</STRONG>
[{b,[1]},{c,[2,3]}]</PRE>
</UL>
<P><A NAME="relative_product%2"><STRONG><CODE>relative_product(TupleOfBinRels [, BinRel1]) -&#62; BinRel2</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>TupleOfBinRels = tuple-of(BinRel)</CODE></STRONG><BR>
<STRONG><CODE>BinRel = BinRel1 = BinRel2 = binary_relation()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>If TupleOfBinRels is a non-empty tuple
         {R[1],&#160;...,&#160;R[n]} of binary relations and BinRel1
         is a binary relation, then BinRel2 is the <A HREF="#tuple_relative_product">relative product</A>
         of the ordered set (R[i],&#160;...,&#160;R[n]) and BinRel1.

<P>If BinRel1 is omitted, the relation of equality between the
elements of the <A HREF="#Cartesian_product_tuple"> Cartesian product</A> of the ranges of R[i],
range&#160;R[1]&#160;&#215;&#160;...&#160;&#215;&#160;range&#160;R[n],
is used instead (intuitively, nothing is &#34;lost&#34;).

        <PRE>1&#62; <STRONG>TR = sofs:relation([{1,a},{1,aa},{2,b}]),</STRONG>
<STRONG>R1 = sofs:relation([{1,u},{2,v},{3,c}]),</STRONG>
<STRONG>R2 = sofs:relative_product({TR, R1}),</STRONG>
<STRONG>sofs:to_external(R2).</STRONG>
[{1,{a,u}},{1,{aa,u}},{2,{b,v}}]</PRE>
<P>Note that <CODE>relative_product({R1},&#160;R2)</CODE> is
         different from <CODE>relative_product(R1,&#160;R2)</CODE>; the
         tuple of one element is not identified with the element
         itself.
</UL>
<P><A NAME="relative_product%2"><STRONG><CODE>relative_product(BinRel1, BinRel2) -&#62; BinRel3</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>BinRel1 = BinRel2 = BinRel3 = binary_relation()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P><A NAME="relprod_impl"><!-- Empty --></A>Returns the <A HREF="#relative_product">relative product</A> of the
         binary relations BinRel1 and BinRel2.
</UL>
<P><A NAME="relative_product1%2"><STRONG><CODE>relative_product1(BinRel1, BinRel2) -&#62; BinRel3</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>BinRel1 = BinRel2 = BinRel3 = binary_relation()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Returns the <A HREF="#relative_product">relative
         product</A> of the <A HREF="#converse">converse</A> of the binary relation
         BinRel1 and the binary relation BinRel2;
         <CODE>relative_product1(R1, R2)</CODE> is equivalent to
         <CODE>relative_product(converse(R1), R2)</CODE>, but is more
         efficient.

        <PRE>1&#62; <STRONG>R1 = sofs:relation([{1,a},{1,aa},{2,b}]),</STRONG>
<STRONG>R2 = sofs:relation([{1,u},{2,v},{3,c}]),</STRONG>
<STRONG>R3 = sofs:relative_product1(R1, R2),</STRONG>
<STRONG>sofs:to_external(R3).</STRONG>
[{a,u},{aa,u},{b,v}]</PRE>
</UL>
<P><A NAME="restriction%2"><STRONG><CODE>restriction(BinRel1, Set) -&#62; BinRel2</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>BinRel1 = BinRel2 = binary_relation()</CODE></STRONG><BR>
<STRONG><CODE>Set = set()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Returns the <A HREF="#restriction">   restriction</A> of the binary relation BinRel1 to Set.

        <PRE>1&#62; <STRONG>R1 = sofs:relation([{1,a},{2,b},{3,c}]),</STRONG>
<STRONG>S = sofs:set([1,2,4]),</STRONG>
<STRONG>R2 = sofs:restriction(R1, S),</STRONG>
<STRONG>sofs:to_external(R2).</STRONG>
[{1,a},{2,b}]</PRE>
</UL>
<P><A NAME="restriction%3"><STRONG><CODE>restriction(SetFun, Set1, Set2) -&#62; Set3</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>SetFun = set_fun()</CODE></STRONG><BR>
<STRONG><CODE>Set1 = Set2 = Set3 = set()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Returns a subset of Set1 containing those elements that
yield an element in Set2 as the result of applying SetFun.

        <PRE>1&#62; <STRONG>S1 = sofs:relation([{1,a},{2,b},{3,c}]),</STRONG>
<STRONG>S2 = sofs:set([b,c,d]),</STRONG>
<STRONG>S3 = sofs:restriction(2, S1, S2),</STRONG>
<STRONG>sofs:to_external(S3).</STRONG>
[{2,b},{3,c}]</PRE>
</UL>
<P><A NAME="set%2"><STRONG><CODE>set(Terms [, Type]) -&#62; Set</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>Set = set()</CODE></STRONG><BR>
<STRONG><CODE>Terms = [term()]</CODE></STRONG><BR>
<STRONG><CODE>Type = type()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Creates an <A HREF="#sets_definition">unordered
         set</A>. <CODE>set(L, T)</CODE> is equivalent to
         <CODE>from_term(L, T)</CODE>, if the result is an unordered set.
         If no <A HREF="#type">type</A> is explicitly
         given, <CODE>[atom]</CODE> is used as type of the set.

</UL>
<P><A NAME="specification%2"><STRONG><CODE>specification(Fun, Set1) -&#62; Set2</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>Fun = spec_fun()</CODE></STRONG><BR>
<STRONG><CODE>Set1 = Set2 = set()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Returns the set containing every element of Set1 for which
         Fun returns <CODE>true</CODE>. If Fun is a tuple <CODE>{external,
         Fun2}</CODE>, Fun2 is applied to the <A HREF="#external_set">external set</A> of each
         element, otherwise Fun is applied to each element.

        <PRE>1&#62; <STRONG>R1 = sofs:relation([{a,1},{b,2}]),</STRONG>
<STRONG>R2 = sofs:relation([{x,1},{x,2},{y,3}]),</STRONG>
<STRONG>S1 = sofs:from_sets([R1,R2]),</STRONG>
<STRONG>S2 = sofs:specification({sofs,is_a_function}, S1),</STRONG>
<STRONG>sofs:to_external(S2).</STRONG>
[[{a,1},{b,2}]]</PRE>
</UL>
<P><A NAME="strict_relation%1"><STRONG><CODE>strict_relation(BinRel1) -&#62; BinRel2</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>BinRel1 = BinRel2 = binary_relation()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Returns the <A HREF="#strict_relation">strict
         relation</A> corresponding to the binary relation
         BinRel1.

        <PRE>1&#62; <STRONG>R1 = sofs:relation([{1,1},{1,2},{2,1},{2,2}]),</STRONG>
<STRONG>R2 = sofs:strict_relation(R1),</STRONG>
<STRONG>sofs:to_external(R2).</STRONG>
[{1,2},{2,1}]</PRE>
</UL>
<P><A NAME="substitution%2"><STRONG><CODE>substitution(SetFun, Set1) -&#62; Set2</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>SetFun = set_fun()</CODE></STRONG><BR>
<STRONG><CODE>Set1 = Set2 = set()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>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.

        <PRE>1&#62; <STRONG>L = [{a,1},{b,2}].</STRONG>
[{a,1},{b,2}]
2&#62; <STRONG>sofs:to_external(sofs:projection(1,sofs:relation(L))).</STRONG>
[a,b]
3&#62; <STRONG>sofs:to_external(sofs:substitution(1,sofs:relation(L))).</STRONG>
[{{a,1},a},{{b,2},b}]
4&#62; <STRONG>SetFun = {external, fun({A,_}=E) -&#62; {E,A} end},</STRONG>
<STRONG>sofs:to_external(sofs:projection(SetFun,sofs:relation(L))).</STRONG>
[{{a,1},a},{{b,2},b}]</PRE>
<P>The relation of equality between the elements of {a,b,c}:

        <PRE>1&#62; <STRONG>I = sofs:substitution(fun(A) -&#62; A end, sofs:set([a,b,c])),</STRONG>
<STRONG>sofs:to_external(I).</STRONG>
[{a,a},{b,b},{c,c}]</PRE>
<P>Let SetOfSets be a set of sets and BinRel a binary
relation. The function that maps each element Set of
SetOfSets onto the <A HREF="#image">image</A>
of Set under BinRel is returned by this function:

<PRE>images(SetOfSets, BinRel) -&#62;
   Fun = fun(Set) -&#62; sofs:image(BinRel, Set) end,
   sofs:substitution(Fun, SetOfSets).</PRE>
<P>Here might be the place to reveal something that was more
or less stated before, namely that external unordered sets
are represented as sorted lists. As a consequence, creating
the image of a set under a relation R may traverse all
elements of R (to that comes the sorting of results, the
image). In <CODE>images/2</CODE>, BinRel will be traversed once
for each element of SetOfSets, which may take too long. The
following efficient function could be used instead, assuming
that SetOfSets does not contain an empty set and that BinRel
is non-empty:

<PRE>images2(SetOfSets, BinRel) -&#62;
   CR = sofs:canonical_relation(SetOfSets),
   R = sofs:relative_product1(CR, BinRel),
   sofs:relation_to_family(R).</PRE>
</UL>
<P><A NAME="symdiff%2"><STRONG><CODE>symdiff(Set1, Set2) -&#62; Set3</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>Set1 = Set2 = Set3 = set()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Returns the <A HREF="#symmetric_difference">  symmetric difference</A> (or the Boolean sum) of Set1
         and Set2.

        <PRE>1&#62; <STRONG>S1 = sofs:set([1,2,3]),</STRONG>
<STRONG>S2 = sofs:set([2,3,4]),</STRONG>
<STRONG>P = sofs:symdiff(S1, S2),</STRONG>
<STRONG>sofs:to_external(P).</STRONG>
[1,4]</PRE>
</UL>
<P><A NAME="symmetric_partition%2"><STRONG><CODE>symmetric_partition(Set1, Set2) -&#62; {Set3, Set4, Set5}</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>Set1 = Set2 = Set3 = Set4 = Set5 = set()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Returns a triple of three 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.
</UL>
<P><A NAME="to_external%1"><STRONG><CODE>to_external(AnySet) -&#62; ExternalSet</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>ExternalSet = external_set()</CODE></STRONG><BR>
<STRONG><CODE>AnySet = anyset()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Returns the <A HREF="#external_set">external
set</A> of an atomic, ordered or unordered set.
</UL>
<P><A NAME="to_sets%1"><STRONG><CODE>to_sets(ASet) -&#62; Sets</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>ASet = set() | ordset()</CODE></STRONG><BR>
<STRONG><CODE>Sets = tuple_of(AnySet) | [AnySet]</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>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.
</UL>
<P><A NAME="type%1"><STRONG><CODE>type(AnySet) -&#62; Type</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>AnySet = anyset()</CODE></STRONG><BR>
<STRONG><CODE>Type = type()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Returns the <A HREF="#type">type</A> of an
         atomic, ordered or unordered set.
</UL>
<P><A NAME="union%1"><STRONG><CODE>union(SetOfSets) -&#62; Set</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>Set = set()</CODE></STRONG><BR>
<STRONG><CODE>SetOfSets = set_of_sets()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Returns the <A HREF="#union_n">union</A> of the
set of sets SetOfSets.
</UL>
<P><A NAME="union%2"><STRONG><CODE>union(Set1, Set2) -&#62; Set3</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>Set1 = Set2 = Set3 = set()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Returns the <A HREF="#union">union</A> of
Set1 and Set2.
</UL>
<P><A NAME="union_of_family%1"><STRONG><CODE>union_of_family(Family) -&#62; Set</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>Family = family()</CODE></STRONG><BR>
<STRONG><CODE>Set = set()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Returns the union of the <A HREF="#family">family</A> Family.

        <PRE>1&#62; <STRONG>F = sofs:family([{a,[0,2,4]},{b,[0,1,2]},{c,[2,3]}]),</STRONG>
<STRONG>S = sofs:union_of_family(F),</STRONG>
<STRONG>sofs:to_external(S).</STRONG>
[0,1,2,3,4]</PRE>
</UL>
<P><A NAME="weak_relation%1"><STRONG><CODE>weak_relation(BinRel1) -&#62; BinRel2</CODE></STRONG></A><BR>
<P><UL>Types:
<UL>
<STRONG><CODE>BinRel1 = BinRel2 = binary_relation()</CODE></STRONG><BR>
</UL>
</UL>
<UL>
<P>Returns a subset S of the <A HREF="#weak_relation">weak relation</A> W
         corresponding to the binary relation BinRel1. Let F be the
         <A HREF="#field">field</A> 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.

        <PRE>1&#62; <STRONG>R1 = sofs:relation([{1,1},{1,2},{3,1}]),</STRONG>
<STRONG>R2 = sofs:weak_relation(R1),</STRONG>
<STRONG>sofs:to_external(R2).</STRONG>
[{1,1},{1,2},{2,2},{3,1},{3,3}]</PRE>
</UL>
<H3>See Also</H3>
<UL>
<P><A HREF="dict.html">dict(3)</A>, 
<A HREF="digraph.html">digraph(3)</A>,
<A HREF="orddict.html">orddict(3)</A>, 
<A HREF="ordsets.html">ordsets(3)</A>, 
<A HREF="sets.html">sets(3)</A>
</UL>
<H3>AUTHORS</H3>
<UL>
Hans Bolinder - support@erlang.ericsson.se<BR>
</UL>
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<FONT SIZE=-1>stdlib 1.10<BR>
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<A HREF="http://www.erlang.se">Ericsson Utvecklings AB</A><BR>
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