path: root/lib/stdlib/doc/src/sofs.xml

<?xml version="1.0" encoding="latin1" ?>
<!DOCTYPE erlref SYSTEM "erlref.dtd">

<erlref>
  <header>
    <copyright>
      <year>2001</year><year>2010</year>
      <holder>Ericsson AB. All Rights Reserved.</holder>
    </copyright>
    <legalnotice>
      The contents of this file are subject to the Erlang Public License,
      Version 1.1, (the "License"); you may not use this file except in
      compliance with the License. You should have received a copy of the
      Erlang Public License along with this software. If not, it can be
      retrieved online at http://www.erlang.org/.

      Software distributed under the License is distributed on an "AS IS"
      basis, WITHOUT WARRANTY OF ANY KIND, either express or implied. See
      the License for the specific language governing rights and limitations
      under the License.

    </legalnotice>

    <title>sofs</title>
    <prepared>Hans Bolinder</prepared>
    <responsible>nobody</responsible>
    <docno></docno>
    <approved>nobody</approved>
    <checked>no</checked>
    <date>2001-08-25</date>
    <rev>PA1</rev>
    <file>sofs.sgml</file>
  </header>
  <module>sofs</module>
  <modulesummary>Functions for Manipulating Sets of Sets</modulesummary>
  <description>
    <p>The <c>sofs</c> 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>
    <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&nbsp;=
      {x&nbsp;in&nbsp;A&nbsp;: 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&nbsp;= {x&nbsp;: S(x)}. </p>
    <p>The <em>unordered set</em> containing the elements a, b and c
      is denoted {a,&nbsp;b,&nbsp;c}. This notation is not to be
      confused with tuples. The <em>ordered pair</em> of a and b, with
      first <em>coordinate</em> a and second coordinate b, is denoted
      (a,&nbsp;b). An ordered pair is an <em>ordered set</em> 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>
    <p>The set that contains no elements is called the <em>empty set</em>.
      If two sets A and B contain the same elements, then A 
      is <marker id="equal"></marker><em>equal</em> to B, denoted
      A&nbsp;=&nbsp;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 <marker id="subset"></marker><em>subset</em> of A. 
      The <marker id="union"></marker><em>union</em> of two sets A and B is 
      the smallest set that contains all elements of A and all elements of
      B. The <marker id="intersection"></marker><em>intersection</em> of two
      sets A and B is the set that contains all elements of A that
      belong to B. 
      Two sets are <marker id="disjoint"></marker><em>disjoint</em> if their
      intersection is the empty set. 
      The <marker id="difference"></marker><em>difference</em> of
      two sets A and B is the set that contains all elements of A that
      do not belong to B. 
      The <marker id="symmetric_difference"></marker><em>symmetric 
      difference</em> of 
      two sets is the set that contains those element that belong to
      either of the two sets, but not both. 
      The <marker id="union_n"></marker><em>union</em> 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 <marker id="intersection_n"></marker><em>intersection</em> of 
      a non-empty collection of sets is the set that contains all elements
      that belong to every set of the collection.</p>
    <p>The <marker id="Cartesian_product"></marker><em>Cartesian 
      product</em> of
      two sets X and Y, denoted X&nbsp;&times;&nbsp;Y, is the set
      {a&nbsp;: a&nbsp;= (x,&nbsp;y) for some x&nbsp;in&nbsp;X and for
      some y&nbsp;in&nbsp;Y}. 
      A <marker id="relation"></marker><em>relation</em> is a subset of
      X&nbsp;&times;&nbsp;Y. Let R be a relation. The fact that
      (x,&nbsp;y) belongs to R is written as x&nbsp;R&nbsp;y. Since
      relations are sets, the definitions of the last paragraph
      (subset, union, and so on) apply to relations as well. 
      The <marker id="domain"></marker><em>domain</em> of R is the 
      set {x&nbsp;: x&nbsp;R&nbsp;y for some y&nbsp;in&nbsp;Y}. 
      The <marker id="range"></marker><em>range</em> of R is the 
      set {y&nbsp;: x&nbsp;R&nbsp;y for some x&nbsp;in&nbsp;X}. 
      The <marker id="converse"></marker><em>converse</em> of R is the 
      set {a&nbsp;: a&nbsp;= (y,&nbsp;x) for some 
      (x,&nbsp;y)&nbsp;in&nbsp;R}. If A is a subset of X, then 
      the <marker id="image"></marker><em>image</em> of
      A under R is the set {y&nbsp;: x&nbsp;R&nbsp;y for some
      x&nbsp;in&nbsp;A}, and if B is a subset of Y, then 
      the <marker id="inverse_image"></marker><em>inverse image</em> of B is 
      the set {x&nbsp;: x&nbsp;R&nbsp;y for some y&nbsp;in&nbsp;B}. If R is a
      relation from X to Y and S is a relation from Y to Z, then 
      the <marker id="relative_product"></marker><em>relative product</em> of
      R and S is the relation T from X to Z defined so that x&nbsp;T&nbsp;z
      if and only if there exists an element y in Y such that
      x&nbsp;R&nbsp;y and y&nbsp;S&nbsp;z. 
      The <marker id="restriction"></marker><em>restriction</em> of R to A is
      the set S defined so that x&nbsp;S&nbsp;y if and only if there exists an
      element x in A such that x&nbsp;R&nbsp;y. If S is a restriction
      of R to A, then R is 
      an <marker id="extension"></marker><em>extension</em> of S to X. 
      If X&nbsp;=&nbsp;Y then we call R a relation <em>in</em> X. 
      The <marker id="field"></marker><em>field</em> 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&nbsp;S&nbsp;y if x&nbsp;R&nbsp;y and
      not x&nbsp;=&nbsp;y, then S is 
      the <marker id="strict_relation"></marker><em>strict</em> relation 
      corresponding to
      R, and vice versa, if S is a relation in X, and if R is defined
      so that x&nbsp;R&nbsp;y if x&nbsp;S&nbsp;y or x&nbsp;=&nbsp;y,
      then R is the <marker id="weak_relation"></marker><em>weak</em> relation
      corresponding to S. A relation R in X is <em>reflexive</em> if
      x&nbsp;R&nbsp;x for every element x of X; it is
      <em>symmetric</em> if x&nbsp;R&nbsp;y implies that
      y&nbsp;R&nbsp;x; and it is <em>transitive</em> if
      x&nbsp;R&nbsp;y and y&nbsp;R&nbsp;z imply that x&nbsp;R&nbsp;z.</p>
    <p>A <marker id="function"></marker><em>function</em> F is a relation, a
      subset of X&nbsp;&times;&nbsp;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,&nbsp;y) in F. The latter condition can
      be formulated as follows: if x&nbsp;F&nbsp;y and x&nbsp;F&nbsp;z
      then y&nbsp;=&nbsp;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,&nbsp;y)&nbsp;in&nbsp;F or x&nbsp;F&nbsp;y, we write
      F(x)&nbsp;=&nbsp;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 <marker id="inverse"></marker><em>inverse</em> of F. 
      The relative product of two functions F1 and F2 is called 
      the <marker id="composite"></marker><em>composite</em> of F1 and F2
      if the range of F1 is a subset of the domain of F2. </p>
    <p>Sometimes, when the range of a function is more important than
      the function itself, the function is called a <em>family</em>.
      The domain of a family is called the <em>index set</em>, and the
      range is called the <em>indexed set</em>. 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 <marker id="family"></marker><em>family of subsets</em> of X. If x
      is a family of subsets of X, then the union of the range of x is
      called the <em>union of the family</em> x. If x is non-empty
      (the index set is non-empty), 
      the <em>intersection of the family</em> 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.</p>
    <p>A <marker id="partition"></marker><em>partition</em> 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
      <em>equivalence relation</em> if it is reflexive, symmetric and
      transitive. If R is an equivalence relation in X, and x is an
      element of X, 
      the <marker id="equivalence_class"></marker><em>equivalence 
      class</em> of x with respect to R is the set of all those
      elements y of X for which x&nbsp;R&nbsp;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 <marker id="canonical_map"></marker><em>canonical map</em> is
      the function that maps every element of X onto its equivalence class.
      </p>
    <p>Relations as defined above (as sets of ordered pairs) will from
      now on be referred to as <em>binary relations</em>. We call a
      set of ordered sets (x[1],&nbsp;...,&nbsp;x[n]) 
      an <em>(n-ary) relation</em>, and say that the relation is a subset of
      the <marker id="Cartesian_product_tuple"></marker>Cartesian product
      X[1]&nbsp;&times;&nbsp;...&nbsp;&times;&nbsp;X[n] where x[i] is
      an element of X[i], 1&nbsp;&lt;=&nbsp;i&nbsp;&lt;=&nbsp;n. 
      The <marker id="projection"></marker><em>projection</em> of an n-ary
      relation R onto coordinate i is the set {x[i]&nbsp;:
      (x[1],&nbsp;...,&nbsp;x[i],&nbsp;...,&nbsp;x[n]) in R for some
      x[j]&nbsp;in&nbsp;X[j], 1&nbsp;&lt;=&nbsp;j&nbsp;&lt;=&nbsp;n
      and not i&nbsp;=&nbsp;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],&nbsp;...,&nbsp;R[n]) of binary
      relations from X to Y[i] and S a binary relation from
      (Y[1]&nbsp;&times;&nbsp;...&nbsp;&times;&nbsp;Y[n]) to Z. 
      The <marker id="tuple_relative_product"></marker><em>relative 
      product</em> of
      TR and S is the binary relation T from X to Z defined so that
      x&nbsp;T&nbsp;z if and only if there exists an element y[i] in
      Y[i] for each 1&nbsp;&lt;=&nbsp;i&nbsp;&lt;=&nbsp;n such that
      x&nbsp;R[i]&nbsp;y[i] and
      (y[1],&nbsp;...,&nbsp;y[n])&nbsp;S&nbsp;z. Now let TR be a an
      ordered set (R[1],&nbsp;...,&nbsp;R[n]) of binary relations from
      X[i] to Y[i] and S a subset of
      X[1]&nbsp;&times;&nbsp;...&nbsp;&times;&nbsp;X[n]. 
      The <marker id="multiple_relative_product"></marker><em>multiple 
      relative product</em> of TR and and S is defined to be the 
      set {z&nbsp;: z&nbsp;= ((x[1],&nbsp;...,&nbsp;x[n]), (y[1],...,y[n]))
      for some (x[1],&nbsp;...,&nbsp;x[n])&nbsp;in&nbsp;S and for some
      (x[i],&nbsp;y[i]) in R[i],
      1&nbsp;&lt;=&nbsp;i&nbsp;&lt;=&nbsp;n}. 
      The <marker id="natural_join"></marker><em>natural join</em> of 
      an n-ary relation R
      and an m-ary relation S on coordinate i and j is defined to be
      the set {z&nbsp;: z&nbsp;= (x[1],&nbsp;...,&nbsp;x[n],&nbsp;
      y[1],&nbsp;...,&nbsp;y[j-1],&nbsp;y[j+1],&nbsp;...,&nbsp;y[m])
      for some (x[1],&nbsp;...,&nbsp;x[n])&nbsp;in&nbsp;R and for some
      (y[1],&nbsp;...,&nbsp;y[m])&nbsp;in&nbsp;S such that
      x[i]&nbsp;=&nbsp;y[j]}.</p>
    <p><marker id="sets_definition"></marker>The sets recognized by this 
      module will be represented by elements of the relation Sets, defined as
      the smallest set such that:</p>
    <list type="bulleted">
      <item>for every atom T except '_' and for every term X,
       (T,&nbsp;X) belongs to Sets (<em>atomic sets</em>);
      </item>
      <item>(['_'],&nbsp;[]) belongs to Sets (the <em>untyped empty set</em>);
      </item>
      <item>for every tuple T&nbsp;= {T[1],&nbsp;...,&nbsp;T[n]} and
       for every tuple X&nbsp;= {X[1],&nbsp;...,&nbsp;X[n]}, if
       (T[i],&nbsp;X[i]) belongs to Sets for every
       1&nbsp;&lt;=&nbsp;i&nbsp;&lt;=&nbsp;n then (T,&nbsp;X) belongs
       to Sets (<em>ordered sets</em>);
      </item>
      <item>for every term T, if X is the empty list or a non-empty
       sorted list [X[1],&nbsp;...,&nbsp;X[n]] without duplicates
       such that (T,&nbsp;X[i]) belongs to Sets for every
       1&nbsp;&lt;=&nbsp;i&nbsp;&lt;=&nbsp;n, then ([T],&nbsp;X)
       belongs to Sets (<em>typed unordered sets</em>).</item>
    </list>
    <p>An <marker id="external_set"></marker><em>external set</em> is an
      element of the range of Sets. 
      A <marker id="type"></marker><em>type</em>
      is an element of the domain of Sets. If S is an element
      (T,&nbsp;X) of Sets, then T is 
      a <marker id="valid_type"></marker><em>valid type</em> of X, 
      T is the type of S, and X is the external set 
      of S. <seealso marker="#from_term">from_term/2</seealso> creates a 
      set from a type and an Erlang term turned into an external set.</p>
    <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>
    <list type="bulleted">
      <item>Set(T,Term)&nbsp;= Term, where T is an atom;</item>
      <item>Set({T[1],&nbsp;...,&nbsp;T[n]},&nbsp;{X[1],&nbsp;...,&nbsp;X[n]})
       &nbsp;= (Set(T[1],&nbsp;X[1]),&nbsp;...,&nbsp;Set(T[n],&nbsp;X[n]));</item>
      <item>Set([T],&nbsp;[X[1],&nbsp;...,&nbsp;X[n]])
       &nbsp;= {Set(T,&nbsp;X[1]),&nbsp;...,&nbsp;Set(T,&nbsp;X[n])};</item>
      <item>Set([T],&nbsp;[])&nbsp;= {}.</item>
    </list>
    <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 <c>union/2</c> 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>
    <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, <seealso marker="#relprod_impl">relative_product/2</seealso>, checks 
      that the arguments represent binary relations by matching [{A,B}]
      against the type of the first argument (Arg1 say), and [{C,D}]
      against the type of the second argument (Arg2 say). The fact
      that [{A,B}] matches the type of Arg1 is to be interpreted as
      Arg1 representing a binary relation from X to Y, where X is
      defined as all sets Set(x) for some element x in Sets the type
      of which is A, and similarly for Y. In the same way Arg2 is
      interpreted as representing a binary relation from W to Z.
      Finally it is checked that B matches C, which is sufficient to
      ensure that W is equal to Y. The untyped empty set is handled
      separately: its type, ['_'], matches the type of any unordered
      set.</p>
    <p>A few functions of this module (<c>drestriction/3</c>,
      <c>family_projection/2</c>, <c>partition/2</c>,
      <c>partition_family/2</c>, <c>projection/2</c>,
      <c>restriction/3</c>, <c>substitution/2</c>) 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
      <c>{external,&nbsp;Fun}</c>, 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 <c>{external, Fun}</c>, 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 <c>{external, fun(X)&nbsp;-> element(I,&nbsp;X)&nbsp;end}</c>,
      but is to be preferred since it makes it possible to handle this
      case even more efficiently. Examples of SetFuns:</p>
    <pre>
{sofs, union}
fun(S) -> sofs:partition(1, S) end
{external, fun(A) -> A end}
{external, fun({A,_,C}) -> {C,A} end}
{external, fun({_,{_,C}}) -> C end}
{external, fun({_,{_,{_,E}=C}}) -> {E,{E,C}} end}
2</pre>
    <p>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.</p>
    <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: <c>from_external</c>,
      <c>is_empty_set</c>, <c>is_set</c>, <c>is_sofs_set</c>,
      <c>to_external</c>, <c>type</c>.</p>
    <p>The functions of this module exit the process with a
      <c>badarg</c>, <c>bad_function</c>, or <c>type_mismatch</c>
      message when given badly formed arguments or sets the types of
      which are not compatible.</p>
    <p>When comparing external sets the operator <c>==/2</c> is used.</p>
    <p><em>Types</em></p>
    <pre>
anyset() = -&nbsp;an unordered, ordered or atomic set&nbsp;-
binary_relation() = -&nbsp;a binary relation&nbsp;-
bool() = true | false
external_set() = -&nbsp;an external set&nbsp;-
family() = -&nbsp;a family (of subsets)&nbsp;-
function() = -&nbsp;a function&nbsp;-
ordset() = -&nbsp;an ordered set&nbsp;-
relation() = -&nbsp;an n-ary relation&nbsp;-
set() = -&nbsp;an unordered set&nbsp;-
set_of_sets() = -&nbsp;an unordered set of set()&nbsp;-
set_fun() = integer() >= 1
          | {external, fun(external_set()) -> external_set()}
          | fun(anyset()) -> anyset()
spec_fun() = {external, fun(external_set()) -> bool()}
           | fun(anyset()) -> bool()
type() = -&nbsp;a type&nbsp;- </pre>
  </description>
  <funcs>
    <func>
      <name>a_function(Tuples [, Type]) -> Function</name>
      <fsummary>Create a function.</fsummary>
      <type>
        <v>Function = function()</v>
        <v>Tuples = [tuple()]</v>
        <v>Type = type()</v>
      </type>
      <desc>
        <p>Creates a <seealso marker="#function">function</seealso>.
          <c>a_function(F,&nbsp;T)</c> is equivalent to
          <c>from_term(F,&nbsp;T)</c>, if the result is a function. If
          no <seealso marker="#type">type</seealso> is explicitly
          given, <c>[{atom,&nbsp;atom}]</c> is used as type of the
          function.</p>
      </desc>
    </func>
    <func>
      <name>canonical_relation(SetOfSets) -> BinRel</name>
      <fsummary>Return the canonical map.</fsummary>
      <type>
        <v>BinRel = binary_relation()</v>
        <v>SetOfSets = set_of_sets()</v>
      </type>
      <desc>
        <p>Returns the binary relation containing the elements
          (E,&nbsp;Set) such that Set belongs to SetOfSets and E
          belongs to Set. If SetOfSets is 
	  a <seealso marker="#partition">partition</seealso> of a set X and
	  R is the equivalence relation in X induced by SetOfSets, then the
          returned relation is 
	  the <seealso marker="#canonical_map">canonical map</seealso> from 
	  X onto the equivalence classes with respect to R.</p>
        <pre>
1> <input>Ss = sofs:from_term([[a,b],[b,c]]),</input>
<input>CR = sofs:canonical_relation(Ss),</input>
<input>sofs:to_external(CR).</input>
[{a,[a,b]},{b,[a,b]},{b,[b,c]},{c,[b,c]}]</pre>
      </desc>
    </func>
    <func>
      <name>composite(Function1, Function2) -> Function3</name>
      <fsummary>Return the composite of two functions.</fsummary>
      <type>
        <v>Function1 = Function2 = Function3 = function()</v>
      </type>
      <desc>
        <p>Returns the <seealso marker="#composite">composite</seealso> of 
	  the functions Function1 and Function2.</p>
        <pre>
1> <input>F1 = sofs:a_function([{a,1},{b,2},{c,2}]),</input>
<input>F2 = sofs:a_function([{1,x},{2,y},{3,z}]),</input>
<input>F = sofs:composite(F1, F2),</input>
<input>sofs:to_external(F).</input>
[{a,x},{b,y},{c,y}]</pre>
      </desc>
    </func>
    <func>
      <name>constant_function(Set, AnySet) -> Function</name>
      <fsummary>Create the function that maps each element of a 
        set onto another set.</fsummary>
      <type>
        <v>AnySet = anyset()</v>
        <v>Function = function()</v>
        <v>Set = set()</v>
      </type>
      <desc>
        <p>Creates the <seealso marker="#function">function</seealso>
          that maps each element of the set Set onto AnySet.</p>
        <pre>
1> <input>S = sofs:set([a,b]),</input>
<input>E = sofs:from_term(1),</input>
<input>R = sofs:constant_function(S, E),</input>
<input>sofs:to_external(R).</input>
[{a,1},{b,1}]</pre>
      </desc>
    </func>
    <func>
      <name>converse(BinRel1) -> BinRel2</name>
      <fsummary>Return the converse of a binary relation.</fsummary>
      <type>
        <v>BinRel1 = BinRel2 = binary_relation()</v>
      </type>
      <desc>
        <p>Returns the <seealso marker="#converse">converse</seealso>
          of the binary relation BinRel1.</p>
        <pre>
1> <input>R1 = sofs:relation([{1,a},{2,b},{3,a}]),</input>
<input>R2 = sofs:converse(R1),</input>
<input>sofs:to_external(R2).</input>
[{a,1},{a,3},{b,2}]</pre>
      </desc>
    </func>
    <func>
      <name>difference(Set1, Set2) -> Set3</name>
      <fsummary>Return the difference of two sets.</fsummary>
      <type>
        <v>Set1 = Set2 = Set3 = set()</v>
      </type>
      <desc>
        <p>Returns the <seealso marker="#difference">difference</seealso> of 
	  the sets Set1 and Set2.</p>
      </desc>
    </func>
    <func>
      <name>digraph_to_family(Graph [, Type]) -> Family</name>
      <fsummary>Create a family from a directed graph.</fsummary>
      <type>
        <v>Graph = digraph() -&nbsp;see digraph(3)&nbsp;-</v>
        <v>Family = family()</v>
        <v>Type = type()</v>
      </type>
      <desc>
        <p>Creates a <seealso marker="#family">family</seealso> from
          the directed graph Graph. Each vertex a of Graph is
          represented by a pair (a,&nbsp;{b[1],&nbsp;...,&nbsp;b[n]})
          where the b[i]'s are the out-neighbours of a. If no type is
          explicitly given, [{atom,&nbsp;[atom]}] is used as type of
          the family. It is assumed that Type is 
	  a <seealso marker="#valid_type">valid type</seealso> of the 
	  external set of the family.</p>
        <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
          <c>family_to_digraph(digraph_to_family(G))</c>.</p>
      </desc>
    </func>
    <func>
      <name>domain(BinRel) -> Set</name>
      <fsummary>Return the domain of a binary relation.</fsummary>
      <type>
        <v>BinRel = binary_relation()</v>
        <v>Set = set()</v>
      </type>
      <desc>
        <p>Returns the <seealso marker="#domain">domain</seealso> of 
	  the binary relation BinRel.</p>
        <pre>
1> <input>R = sofs:relation([{1,a},{1,b},{2,b},{2,c}]),</input>
<input>S = sofs:domain(R),</input>
<input>sofs:to_external(S).</input>
[1,2]</pre>
      </desc>
    </func>
    <func>
      <name>drestriction(BinRel1, Set) -> BinRel2</name>
      <fsummary>Return a restriction of a binary relation.</fsummary>
      <type>
        <v>BinRel1 = BinRel2 = binary_relation()</v>
        <v>Set = set()</v>
      </type>
      <desc>
        <p>Returns the difference between the binary relation BinRel1
          and the <seealso marker="#restriction">restriction</seealso>
          of BinRel1 to Set.</p>
        <pre>
1> <input>R1 = sofs:relation([{1,a},{2,b},{3,c}]),</input>
<input>S = sofs:set([2,4,6]),</input>
<input>R2 = sofs:drestriction(R1, S),</input>
<input>sofs:to_external(R2).</input>
[{1,a},{3,c}]</pre>
        <p><c>drestriction(R,&nbsp;S)</c> is equivalent to
          <c>difference(R,&nbsp;restriction(R,&nbsp;S))</c>.</p>
      </desc>
    </func>
    <func>
      <name>drestriction(SetFun, Set1, Set2) -> Set3</name>
      <fsummary>Return a restriction of a relation.</fsummary>
      <type>
        <v>SetFun = set_fun()</v>
        <v>Set1 = Set2 = Set3 = set()</v>
      </type>
      <desc>
        <p>Returns a subset of Set1 containing those elements that do
          not yield an element in Set2 as the result of applying
          SetFun.</p>
        <pre>
1> <input>SetFun = {external, fun({_A,B,C}) -> {B,C} end},</input>
<input>R1 = sofs:relation([{a,aa,1},{b,bb,2},{c,cc,3}]),</input>
<input>R2 = sofs:relation([{bb,2},{cc,3},{dd,4}]),</input>
<input>R3 = sofs:drestriction(SetFun, R1, R2),</input>
<input>sofs:to_external(R3).</input>
[{a,aa,1}]</pre>
        <p><c>drestriction(F,&nbsp;S1,&nbsp;S2)</c> is equivalent to
          <c>difference(S1,&nbsp;restriction(F,&nbsp;S1,&nbsp;S2))</c>.</p>
      </desc>
    </func>
    <func>
      <name>empty_set() -> Set</name>
      <fsummary>Return the untyped empty set.</fsummary>
      <type>
        <v>Set = set()</v>
      </type>
      <desc>
        <p>Returns the <seealso marker="#sets_definition">untyped empty 
          set</seealso>. <c>empty_set()</c> is equivalent to
          <c>from_term([],&nbsp;['_'])</c>.</p>
      </desc>
    </func>
    <func>
      <name>extension(BinRel1, Set, AnySet) -> BinRel2</name>
      <fsummary>Extend the domain of a binary relation.</fsummary>
      <type>
        <v>AnySet = anyset()</v>
        <v>BinRel1 = BinRel2 = binary_relation()</v>
        <v>Set = set()</v>
      </type>
      <desc>
        <p>Returns the <seealso marker="#extension">extension</seealso> of 
	  BinRel1 such that
          for each element E in Set that does not belong to the
          <seealso marker="#domain">domain</seealso> of BinRel1,
          BinRel2 contains the pair (E,&nbsp;AnySet).</p>
        <pre>
1> <input>S = sofs:set([b,c]),</input>
<input>A = sofs:empty_set(),</input>
<input>R = sofs:family([{a,[1,2]},{b,[3]}]),</input>
<input>X = sofs:extension(R, S, A),</input>
<input>sofs:to_external(X).</input>
[{a,[1,2]},{b,[3]},{c,[]}]</pre>
      </desc>
    </func>
    <func>
      <name>family(Tuples [, Type]) -> Family</name>
      <fsummary>Create a family of subsets.</fsummary>
      <type>
        <v>Family = family()</v>
        <v>Tuples = [tuple()]</v>
        <v>Type = type()</v>
      </type>
      <desc>
        <p>Creates a <seealso marker="#family">family of subsets</seealso>. 
	  <c>family(F,&nbsp;T)</c> is equivalent to
          <c>from_term(F,&nbsp;T)</c>, if the result is a family. If
          no <seealso marker="#type">type</seealso> is explicitly
          given, <c>[{atom,&nbsp;[atom]}]</c> is used as type of the
          family.</p>
      </desc>
    </func>
    <func>
      <name>family_difference(Family1, Family2) -> Family3</name>
      <fsummary>Return the difference of two families.</fsummary>
      <type>
        <v>Family1 = Family2 = Family3 = family()</v>
      </type>
      <desc>
        <p>If Family1 and Family2 
	  are <seealso marker="#family">families</seealso>, 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.</p>
        <pre>
1> <input>F1 = sofs:family([{a,[1,2]},{b,[3,4]}]),</input>
<input>F2 = sofs:family([{b,[4,5]},{c,[6,7]}]),</input>
<input>F3 = sofs:family_difference(F1, F2),</input>
<input>sofs:to_external(F3).</input>
[{a,[1,2]},{b,[3]}]</pre>
      </desc>
    </func>
    <func>
      <name>family_domain(Family1) -> Family2</name>
      <fsummary>Return a family of domains.</fsummary>
      <type>
        <v>Family1 = Family2 = family()</v>
      </type>
      <desc>
        <p>If Family1 is a <seealso marker="#family">family</seealso>
          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 <seealso marker="#domain">domain</seealso> of Family1[i].</p>
        <pre>
1> <input>FR = sofs:from_term([{a,[{1,a},{2,b},{3,c}]},{b,[]},{c,[{4,d},{5,e}]}]),</input>
<input>F = sofs:family_domain(FR),</input>
<input>sofs:to_external(F).</input>
[{a,[1,2,3]},{b,[]},{c,[4,5]}]</pre>
      </desc>
    </func>
    <func>
      <name>family_field(Family1) -> Family2</name>
      <fsummary>Return a family of fields.</fsummary>
      <type>
        <v>Family1 = Family2 = family()</v>
      </type>
      <desc>
        <p>If Family1 is a <seealso marker="#family">family</seealso>
          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 <seealso marker="#field">field</seealso> of Family1[i].</p>
        <pre>
1> <input>FR = sofs:from_term([{a,[{1,a},{2,b},{3,c}]},{b,[]},{c,[{4,d},{5,e}]}]),</input>
<input>F = sofs:family_field(FR),</input>
<input>sofs:to_external(F).</input>
[{a,[1,2,3,a,b,c]},{b,[]},{c,[4,5,d,e]}]</pre>
        <p><c>family_field(Family1)</c> is equivalent to
          <c>family_union(family_domain(Family1), family_range(Family1))</c>.</p>
      </desc>
    </func>
    <func>
      <name>family_intersection(Family1) -> Family2</name>
      <fsummary>Return the intersection of a family
         of sets of sets.</fsummary>
      <type>
        <v>Family1 = Family2 = family()</v>
      </type>
      <desc>
        <p>If Family1 is a <seealso marker="#family">family</seealso>
          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 <seealso marker="#intersection_n">intersection</seealso> of
          Family1[i].</p>
        <p>If Family1[i] is an empty set for some i, then the process
          exits with a <c>badarg</c> message.</p>
        <pre>
1> <input>F1 = sofs:from_term([{a,[[1,2,3],[2,3,4]]},{b,[[x,y,z],[x,y]]}]),</input>
<input>F2 = sofs:family_intersection(F1),</input>
<input>sofs:to_external(F2).</input>
[{a,[2,3]},{b,[x,y]}]</pre>
      </desc>
    </func>
    <func>
      <name>family_intersection(Family1, Family2) -> Family3</name>
      <fsummary>Return the intersection of two families.</fsummary>
      <type>
        <v>Family1 = Family2 = Family3 = family()</v>
      </type>
      <desc>
        <p>If Family1 and Family2 
	  are <seealso marker="#family">families</seealso>, 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].</p>
        <pre>
1> <input>F1 = sofs:family([{a,[1,2]},{b,[3,4]},{c,[5,6]}]),</input>
<input>F2 = sofs:family([{b,[4,5]},{c,[7,8]},{d,[9,10]}]),</input>
<input>F3 = sofs:family_intersection(F1, F2),</input>
<input>sofs:to_external(F3).</input>
[{b,[4]},{c,[]}]</pre>
      </desc>
    </func>
    <func>
      <name>family_projection(SetFun, Family1) -> Family2</name>
      <fsummary>Return a family of modified subsets.</fsummary>
      <type>
        <v>SetFun = set_fun()</v>
        <v>Family1 = Family2 = family()</v>
        <v>Set = set()</v>
      </type>
      <desc>
        <p>If Family1 is a <seealso marker="#family">family</seealso>
          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.</p>
        <pre>
1> <input>F1 = sofs:from_term([{a,[[1,2],[2,3]]},{b,[[]]}]),</input>
<input>F2 = sofs:family_projection({sofs, union}, F1),</input>
<input>sofs:to_external(F2).</input>
[{a,[1,2,3]},{b,[]}]</pre>
      </desc>
    </func>
    <func>
      <name>family_range(Family1) -> Family2</name>
      <fsummary>Return a family of ranges.</fsummary>
      <type>
        <v>Family1 = Family2 = family()</v>
      </type>
      <desc>
        <p>If Family1 is a <seealso marker="#family">family</seealso>
          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 <seealso marker="#range">range</seealso> of Family1[i].</p>
        <pre>
1> <input>FR = sofs:from_term([{a,[{1,a},{2,b},{3,c}]},{b,[]},{c,[{4,d},{5,e}]}]),</input>
<input>F = sofs:family_range(FR),</input>
<input>sofs:to_external(F).</input>
[{a,[a,b,c]},{b,[]},{c,[d,e]}]</pre>
      </desc>
    </func>
    <func>
      <name>family_specification(Fun, Family1) -> Family2</name>
      <fsummary>Select a subset of a family using a predicate.</fsummary>
      <type>
        <v>Fun = spec_fun()</v>
        <v>Family1 = Family2 = family()</v>
      </type>
      <desc>
        <p>If Family1 is a <seealso marker="#family">family</seealso>,
          then Family2 is 
	  the <seealso marker="#restriction">restriction</seealso> of 
	  Family1 to those elements i of the
          index set for which Fun applied to Family1[i] returns
          <c>true</c>. If Fun is a tuple <c>{external,&nbsp;Fun2}</c>,
          Fun2 is applied to 
	  the <seealso marker="#external_set">external set</seealso> of 
	  Family1[i], otherwise Fun is applied to Family1[i].</p>
        <pre>
1> <input>F1 = sofs:family([{a,[1,2,3]},{b,[1,2]},{c,[1]}]),</input>
<input>SpecFun = fun(S) -> sofs:no_elements(S) =:= 2 end,</input>
<input>F2 = sofs:family_specification(SpecFun, F1),</input>
<input>sofs:to_external(F2).</input>
[{b,[1,2]}]</pre>
      </desc>
    </func>
    <func>
      <name>family_to_digraph(Family [, GraphType]) -> Graph</name>
      <fsummary>Create a directed graph from a family.</fsummary>
      <type>
        <v>Graph = digraph()</v>
        <v>Family = family()</v>
        <v>GraphType = -&nbsp;see digraph(3)&nbsp;-</v>
      </type>
      <desc>
        <p>Creates a directed graph from 
	  the <seealso marker="#family">family</seealso> Family. For each 
	  pair (a,&nbsp;{b[1],&nbsp;...,&nbsp;b[n]}) of Family, the vertex
          a as well the edges (a,&nbsp;b[i]) for
          1&nbsp;&lt;=&nbsp;i&nbsp;&lt;=&nbsp;n are added to a newly
          created directed graph.</p>
        <p>If no graph type is given, <c>digraph:new/1</c> is used for
          creating the directed graph, otherwise the GraphType
          argument is passed on as second argument to
          <c>digraph:new/2</c>.</p>
        <p>It F is a family, it holds that F is a subset of
          <c>digraph_to_family(family_to_digraph(F),&nbsp;type(F))</c>.
          Equality holds if <c>union_of_family(F)</c> is a subset of
          <c>domain(F)</c>.</p>
        <p>Creating a cycle in an acyclic graph exits the process with
          a <c>cyclic</c> message.</p>
      </desc>
    </func>
    <func>
      <name>family_to_relation(Family) -> BinRel</name>
      <fsummary>Create a binary relation from a family.</fsummary>
      <type>
        <v>Family = family()</v>
        <v>BinRel = binary_relation()</v>
      </type>
      <desc>
        <p>If Family is a <seealso marker="#family">family</seealso>,
          then BinRel is the binary relation containing all pairs
          (i,&nbsp;x) such that i belongs to the index set of Family
          and x belongs to Family[i].</p>
        <pre>
1> <input>F = sofs:family([{a,[]}, {b,[1]}, {c,[2,3]}]),</input>
<input>R = sofs:family_to_relation(F),</input>
<input>sofs:to_external(R).</input>
[{b,1},{c,2},{c,3}]</pre>
      </desc>
    </func>
    <func>
      <name>family_union(Family1) -> Family2</name>
      <fsummary>Return the union of a family of sets of sets.</fsummary>
      <type>
        <v>Family1 = Family2 = family()</v>
      </type>
      <desc>
        <p>If Family1 is a <seealso marker="#family">family</seealso>
          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 <seealso marker="#union_n">union</seealso> of Family1[i].</p>
        <pre>
1> <input>F1 = sofs:from_term([{a,[[1,2],[2,3]]},{b,[[]]}]),</input>
<input>F2 = sofs:family_union(F1),</input>
<input>sofs:to_external(F2).</input>
[{a,[1,2,3]},{b,[]}]</pre>
        <p><c>family_union(F)</c> is equivalent to
          <c>family_projection({sofs,union},&nbsp;F)</c>.</p>
      </desc>
    </func>
    <func>
      <name>family_union(Family1, Family2) -> Family3</name>
      <fsummary>Return the union of two families.</fsummary>
      <type>
        <v>Family1 = Family2 = Family3 = family()</v>
      </type>
      <desc>
        <p>If Family1 and Family2 
	  are <seealso marker="#family">families</seealso>, 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.</p>
        <pre>
1> <input>F1 = sofs:family([{a,[1,2]},{b,[3,4]},{c,[5,6]}]),</input>
<input>F2 = sofs:family([{b,[4,5]},{c,[7,8]},{d,[9,10]}]),</input>
<input>F3 = sofs:family_union(F1, F2),</input>
<input>sofs:to_external(F3).</input>
[{a,[1,2]},{b,[3,4,5]},{c,[5,6,7,8]},{d,[9,10]}]</pre>
      </desc>
    </func>
    <func>
      <name>field(BinRel) -> Set</name>
      <fsummary>Return the field of a binary relation.</fsummary>
      <type>
        <v>BinRel = binary_relation()</v>
        <v>Set = set()</v>
      </type>
      <desc>
        <p>Returns the <seealso marker="#field">field</seealso> of the
          binary relation BinRel.</p>
        <pre>
1> <input>R = sofs:relation([{1,a},{1,b},{2,b},{2,c}]),</input>
<input>S = sofs:field(R),</input>
<input>sofs:to_external(S).</input>
[1,2,a,b,c]</pre>
        <p><c>field(R)</c> is equivalent 
	  to <c>union(domain(R), range(R))</c>.</p>
      </desc>
    </func>
    <func>
      <name>from_external(ExternalSet, Type) -> AnySet</name>
      <fsummary>Create a set.</fsummary>
      <type>
        <v>ExternalSet = external_set()</v>
        <v>AnySet = anyset()</v>
        <v>Type = type()</v>
      </type>
      <desc>
        <p>Creates a set from the <seealso marker="#external_set">external 
	  set</seealso> ExternalSet
          and the <seealso marker="#type">type</seealso> Type. It is
          assumed that Type is a <seealso marker="#valid_type">valid 
	  type</seealso> of ExternalSet.</p>
      </desc>
    </func>
    <func>
      <name>from_sets(ListOfSets) -> Set</name>
      <fsummary>Create a set out of a list of sets.</fsummary>
      <type>
        <v>Set = set()</v>
        <v>ListOfSets = [anyset()]</v>
      </type>
      <desc>
        <p>Returns the <seealso marker="#sets_definition">unordered 
	  set</seealso> containing the sets of the list ListOfSets.</p>
        <pre>
1> <input>S1 = sofs:relation([{a,1},{b,2}]),</input>
<input>S2 = sofs:relation([{x,3},{y,4}]),</input>
<input>S = sofs:from_sets([S1,S2]),</input>
<input>sofs:to_external(S).</input>
[[{a,1},{b,2}],[{x,3},{y,4}]]</pre>
      </desc>
    </func>
    <func>
      <name>from_sets(TupleOfSets) -> Ordset</name>
      <fsummary>Create an ordered set out of a tuple of sets.</fsummary>
      <type>
        <v>Ordset = ordset()</v>
        <v>TupleOfSets = tuple-of(anyset())</v>
      </type>
      <desc>
        <p>Returns the <seealso marker="#sets_definition">ordered 
	  set</seealso> containing the sets of the non-empty tuple 
	  TupleOfSets.</p>
      </desc>
    </func>
    <func>
      <name>from_term(Term [, Type]) -> AnySet</name>
      <fsummary>Create a set.</fsummary>
      <type>
        <v>AnySet = anyset()</v>
        <v>Term = term()</v>
        <v>Type = type()</v>
      </type>
      <desc>
        <p><marker id="from_term"></marker>Creates an element 
	  of <seealso marker="#sets_definition">Sets</seealso> by 
	  traversing the term Term, sorting lists, removing duplicates and
	  deriving or verifying a <seealso marker="#valid_type">valid 
	  type</seealso> for the so obtained external set. An
          explicitly given <seealso marker="#type">type</seealso> 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:</p>
        <pre>
1> <input>S = sofs:from_term([{{"foo"},[1,1]},{"foo",[2,2]}], [{atom,[atom]}]),</input>
<input>sofs:to_external(S).</input>
[{{"foo"},[1]},{"foo",[2]}]</pre>
        <p><c>from_term</c> 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 <em>do</em> anything operate on unordered sets.
          Creating unordered sets from a collection of ordered sets
          may be the way to go if the ordered sets are big and one
          does not want to waste heap by rebuilding the elements of
          the unordered set. An example showing that a set can be
          built "layer by layer":</p>
        <pre>
1> <input>A = sofs:from_term(a),</input>
<input>S = sofs:set([1,2,3]),</input>
<input>P1 = sofs:from_sets({A,S}),</input>
<input>P2 = sofs:from_term({b,[6,5,4]}),</input>
<input>Ss = sofs:from_sets([P1,P2]),</input>
<input>sofs:to_external(Ss).</input>
[{a,[1,2,3]},{b,[4,5,6]}]</pre>
        <p>Other functions that create sets are <c>from_external/2</c>
          and <c>from_sets/1</c>. Special cases of <c>from_term/2</c>
          are <c>a_function/1,2</c>, <c>empty_set/0</c>,
          <c>family/1,2</c>, <c>relation/1,2</c>, and <c>set/1,2</c>.</p>
      </desc>
    </func>
    <func>
      <name>image(BinRel, Set1) -> Set2</name>
      <fsummary>Return the image of a set under a binary relation.</fsummary>
      <type>
        <v>BinRel = binary_relation()</v>
        <v>Set1 = Set2 = set()</v>
      </type>
      <desc>
        <p>Returns the <seealso marker="#image">image</seealso> of the
          set Set1 under the binary relation BinRel.</p>
        <pre>
1> <input>R = sofs:relation([{1,a},{2,b},{2,c},{3,d}]),</input>
<input>S1 = sofs:set([1,2]),</input>
<input>S2 = sofs:image(R, S1),</input>
<input>sofs:to_external(S2).</input>
[a,b,c]</pre>
      </desc>
    </func>
    <func>
      <name>intersection(SetOfSets) -> Set</name>
      <fsummary>Return the intersection of a set of sets.</fsummary>
      <type>
        <v>Set = set()</v>
        <v>SetOfSets = set_of_sets()</v>
      </type>
      <desc>
        <p>Returns 
	  the <seealso marker="#intersection_n">intersection</seealso> of 
	  the set of sets SetOfSets.</p>
        <p>Intersecting an empty set of sets exits the process with a
          <c>badarg</c> message.</p>
      </desc>
    </func>
    <func>
      <name>intersection(Set1, Set2) -> Set3</name>
      <fsummary>Return the intersection of two sets.</fsummary>
      <type>
        <v>Set1 = Set2 = Set3 = set()</v>
      </type>
      <desc>
        <p>Returns 
	  the <seealso marker="#intersection">intersection</seealso> of 
	  Set1 and Set2.</p>
      </desc>
    </func>
    <func>
      <name>intersection_of_family(Family) -> Set</name>
      <fsummary>Return the intersection of a family.</fsummary>
      <type>
        <v>Family = family()</v>
        <v>Set = set()</v>
      </type>
      <desc>
        <p>Returns the intersection of 
	  the <seealso marker="#family">family</seealso> Family.</p>
        <p>Intersecting an empty family exits the process with a
          <c>badarg</c> message.</p>
        <pre>
1> <input>F = sofs:family([{a,[0,2,4]},{b,[0,1,2]},{c,[2,3]}]),</input>
<input>S = sofs:intersection_of_family(F),</input>
<input>sofs:to_external(S).</input>
[2]</pre>
      </desc>
    </func>
    <func>
      <name>inverse(Function1) -> Function2</name>
      <fsummary>Return the inverse of a function.</fsummary>
      <type>
        <v>Function1 = Function2 = function()</v>
      </type>
      <desc>
        <p>Returns the <seealso marker="#inverse">inverse</seealso>
          of the function Function1.</p>
        <pre>
1> <input>R1 = sofs:relation([{1,a},{2,b},{3,c}]),</input>
<input>R2 = sofs:inverse(R1),</input>
<input>sofs:to_external(R2).</input>
[{a,1},{b,2},{c,3}]</pre>
      </desc>
    </func>
    <func>
      <name>inverse_image(BinRel, Set1) -> Set2</name>
      <fsummary>Return the inverse image of a set under 
        a binary relation.</fsummary>
      <type>
        <v>BinRel = binary_relation()</v>
        <v>Set1 = Set2 = set()</v>
      </type>
      <desc>
        <p>Returns the <seealso marker="#inverse_image">inverse 
	  image</seealso> of Set1 under the binary relation BinRel.</p>
        <pre>
1> <input>R = sofs:relation([{1,a},{2,b},{2,c},{3,d}]),</input>
<input>S1 = sofs:set([c,d,e]),</input>
<input>S2 = sofs:inverse_image(R, S1),</input>
<input>sofs:to_external(S2).</input>
[2,3]</pre>
      </desc>
    </func>
    <func>
      <name>is_a_function(BinRel) -> Bool</name>
      <fsummary>Test for a function.</fsummary>
      <type>
        <v>Bool = bool()</v>
        <v>BinRel = binary_relation()</v>
      </type>
      <desc>
        <p>Returns <c>true</c> if the binary relation BinRel is a
          <seealso marker="#function">function</seealso> or the
          untyped empty set, <c>false</c> otherwise.</p>
      </desc>
    </func>
    <func>
      <name>is_disjoint(Set1, Set2) -> Bool</name>
      <fsummary>Test for disjoint sets.</fsummary>
      <type>
        <v>Bool = bool()</v>
        <v>Set1 = Set2 = set()</v>
      </type>
      <desc>
        <p>Returns <c>true</c> if Set1 and Set2 
	  are <seealso marker="#disjoint">disjoint</seealso>, <c>false</c>
          otherwise.</p>
      </desc>
    </func>
    <func>
      <name>is_empty_set(AnySet) -> Bool</name>
      <fsummary>Test for an empty set.</fsummary>
      <type>
        <v>AnySet = anyset()</v>
        <v>Bool = bool()</v>
      </type>
      <desc>
        <p>Returns <c>true</c> if Set is an empty unordered set,
          <c>false</c> otherwise.</p>
      </desc>
    </func>
    <func>
      <name>is_equal(AnySet1, AnySet2) -> Bool</name>
      <fsummary>Test two sets for equality.</fsummary>
      <type>
        <v>AnySet1 = AnySet2 = anyset()</v>
        <v>Bool = bool()</v>
      </type>
      <desc>
        <p>Returns <c>true</c> if the AnySet1 and AnySet2 
	  are <seealso marker="#equal">equal</seealso>, <c>false</c> 
	  otherwise. This example shows that <c>==/2</c> is used when
          comparing sets for equality:</p>
        <pre>
1> <input>S1 = sofs:set([1.0]),</input>
<input>S2 = sofs:set([1]),</input>
<input>sofs:is_equal(S1, S2).</input>
true</pre>
      </desc>
    </func>
    <func>
      <name>is_set(AnySet) -> Bool</name>
      <fsummary>Test for an unordered set.</fsummary>
      <type>
        <v>AnySet = anyset()</v>
        <v>Bool = bool()</v>
      </type>
      <desc>
        <p>Returns <c>true</c> if AnySet is 
	  an <seealso marker="#sets_definition">unordered set</seealso>, and
          <c>false</c> if AnySet is an ordered set or an atomic set.</p>
      </desc>
    </func>
    <func>
      <name>is_sofs_set(Term) -> Bool</name>
      <fsummary>Test for an unordered set.</fsummary>
      <type>
        <v>Bool = bool()</v>
        <v>Term = term()</v>
      </type>
      <desc>
        <p>Returns <c>true</c> if Term is 
          an <seealso marker="#sets_definition">unordered set</seealso>, an
          ordered set or an atomic set, <c>false</c> otherwise.</p>
      </desc>
    </func>
    <func>
      <name>is_subset(Set1, Set2) -> Bool</name>
      <fsummary>Test two sets for subset.</fsummary>
      <type>
        <v>Bool = bool()</v>
        <v>Set1 = Set2 = set()</v>
      </type>
      <desc>
        <p>Returns <c>true</c> if Set1 is 
	  a <seealso marker="#subset">subset</seealso> of Set2, <c>false</c>
          otherwise.</p>
      </desc>
    </func>
    <func>
      <name>is_type(Term) -> Bool</name>
      <fsummary>Test for a type.</fsummary>
      <type>
        <v>Bool = bool()</v>
        <v>Term = term()</v>
      </type>
      <desc>
        <p>Returns <c>true</c> if the term Term is 
	  a <seealso marker="#type">type</seealso>.</p>
      </desc>
    </func>
    <func>
      <name>join(Relation1, I, Relation2, J) -> Relation3</name>
      <fsummary>Return the join of two relations.</fsummary>
      <type>
        <v>Relation1 = Relation2 = Relation3 = relation()</v>
        <v>I = J = integer() > 0</v>
      </type>
      <desc>
        <p>Returns the <seealso marker="#natural_join">natural 
	  join</seealso> of the relations Relation1 and Relation2 on
          coordinates I and J.</p>
        <pre>
1> <input>R1 = sofs:relation([{a,x,1},{b,y,2}]),</input>
<input>R2 = sofs:relation([{1,f,g},{1,h,i},{2,3,4}]),</input>
<input>J = sofs:join(R1, 3, R2, 1),</input>
<input>sofs:to_external(J).</input>
[{a,x,1,f,g},{a,x,1,h,i},{b,y,2,3,4}]</pre>
      </desc>
    </func>
    <func>
      <name>multiple_relative_product(TupleOfBinRels, BinRel1) -> BinRel2</name>
      <fsummary>Return the multiple relative product of a tuple of binary 
        relations and a relation.</fsummary>
      <type>
        <v>TupleOfBinRels = tuple-of(BinRel)</v>
        <v>BinRel = BinRel1 = BinRel2 = binary_relation()</v>
      </type>
      <desc>
        <p>If TupleOfBinRels is a non-empty tuple
          {R[1],&nbsp;...,&nbsp;R[n]} of binary relations and BinRel1
          is a binary relation, then BinRel2 is 
	  the <seealso marker="#multiple_relative_product">multiple relative 
	  product</seealso> of the ordered set
          (R[i],&nbsp;...,&nbsp;R[n]) and BinRel1.</p>
        <pre>
1> <input>Ri = sofs:relation([{a,1},{b,2},{c,3}]),</input>
<input>R = sofs:relation([{a,b},{b,c},{c,a}]),</input>
<input>MP = sofs:multiple_relative_product({Ri, Ri}, R),</input>
<input>sofs:to_external(sofs:range(MP)).</input>
[{1,2},{2,3},{3,1}]</pre>
      </desc>
    </func>
    <func>
      <name>no_elements(ASet) -> NoElements</name>
      <fsummary>Return the number of elements of a set.</fsummary>
      <type>
        <v>ASet = set() | ordset()</v>
        <v>NoElements = integer() >= 0 </v>
      </type>
      <desc>
        <p>Returns the number of elements of the ordered or unordered
          set ASet.</p>
      </desc>
    </func>
    <func>
      <name>partition(SetOfSets) -> Partition</name>
      <fsummary>Return the coarsest partition given a set of sets.</fsummary>
      <type>
        <v>SetOfSets = set_of_sets()</v>
        <v>Partition = set()</v>
      </type>
      <desc>
        <p>Returns the <seealso marker="#partition">partition</seealso> 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.</p>
        <pre>
1> <input>Sets1 = sofs:from_term([[a,b,c],[d,e,f],[g,h,i]]),</input>
<input>Sets2 = sofs:from_term([[b,c,d],[e,f,g],[h,i,j]]),</input>
<input>P = sofs:partition(sofs:union(Sets1, Sets2)),</input>
<input>sofs:to_external(P).</input>
[[a],[b,c],[d],[e,f],[g],[h,i],[j]]</pre>
      </desc>
    </func>
    <func>
      <name>partition(SetFun, Set) -> Partition</name>
      <fsummary>Return a partition of a set.</fsummary>
      <type>
        <v>SetFun = set_fun()</v>
        <v>Partition = set()</v>
        <v>Set = set()</v>
      </type>
      <desc>
        <p>Returns the <seealso marker="#partition">partition</seealso> of 
          Set such that two elements are considered equal if the
          results of applying SetFun are equal.</p>
        <pre>
1> <input>Ss = sofs:from_term([[a],[b],[c,d],[e,f]]),</input>
<input>SetFun = fun(S) -> sofs:from_term(sofs:no_elements(S)) end,</input>
<input>P = sofs:partition(SetFun, Ss),</input>
<input>sofs:to_external(P).</input>
[[[a],[b]],[[c,d],[e,f]]]</pre>
      </desc>
    </func>
    <func>
      <name>partition(SetFun, Set1, Set2) -> {Set3, Set4}</name>
      <fsummary>Return a partition of a set.</fsummary>
      <type>
        <v>SetFun = set_fun()</v>
        <v>Set1 = Set2 = Set3 = Set4 = set()</v>
      </type>
      <desc>
        <p>Returns a pair of sets that, regarded as constituting a
          set, forms a <seealso marker="#partition">partition</seealso> 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.</p>
        <pre>
1> <input>R1 = sofs:relation([{1,a},{2,b},{3,c}]),</input>
<input>S = sofs:set([2,4,6]),</input>
<input>{R2,R3} = sofs:partition(1, R1, S),</input>
<input>{sofs:to_external(R2),sofs:to_external(R3)}.</input>
{[{2,b}],[{1,a},{3,c}]}</pre>
        <p><c>partition(F,&nbsp;S1,&nbsp;S2)</c> is equivalent to
          <c>{restriction(F,&nbsp;S1,&nbsp;S2), 
          drestriction(F,&nbsp;S1,&nbsp;S2)}</c>.</p>
      </desc>
    </func>
    <func>
      <name>partition_family(SetFun, Set) -> Family</name>
      <fsummary>Return a family indexing a partition.</fsummary>
      <type>
        <v>Family = family()</v>
        <v>SetFun = set_fun()</v>
        <v>Set = set()</v>
      </type>
      <desc>
        <p>Returns the <seealso marker="#family">family</seealso>
          Family where the indexed set is 
	  a <seealso marker="#partition">partition</seealso> 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 <seealso marker="#equivalence_class">equivalence 
	  class</seealso>.</p>
        <pre>
1> <input>S = sofs:relation([{a,a,a,a},{a,a,b,b},{a,b,b,b}]),</input>
<input>SetFun = {external, fun({A,_,C,_}) -> {A,C} end},</input>
<input>F = sofs:partition_family(SetFun, S),</input>
<input>sofs:to_external(F).</input>
[{{a,a},[{a,a,a,a}]},{{a,b},[{a,a,b,b},{a,b,b,b}]}]</pre>
      </desc>
    </func>
    <func>
      <name>product(TupleOfSets) -> Relation</name>
      <fsummary>Return the Cartesian product of a tuple of sets.</fsummary>
      <type>
        <v>Relation = relation()</v>
        <v>TupleOfSets = tuple-of(set())</v>
      </type>
      <desc>
        <p>Returns the <seealso marker="#Cartesian_product_tuple">Cartesian 
	  product</seealso> of the non-empty tuple of sets
          TupleOfSets. If (x[1],&nbsp;...,&nbsp;x[n]) is an element of
          the n-ary relation Relation, then x[i] is drawn from element
          i of TupleOfSets.</p>
        <pre>
1> <input>S1 = sofs:set([a,b]),</input>
<input>S2 = sofs:set([1,2]),</input>
<input>S3 = sofs:set([x,y]),</input>
<input>P3 = sofs:product({S1,S2,S3}),</input>
<input>sofs:to_external(P3).</input>
[{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>
      </desc>
    </func>
    <func>
      <name>product(Set1, Set2) -> BinRel</name>
      <fsummary>Return the Cartesian product of two sets.</fsummary>
      <type>
        <v>BinRel = binary_relation()</v>
        <v>Set1 = Set2 = set()</v>
      </type>
      <desc>
        <p>Returns the <seealso marker="#Cartesian_product">Cartesian 
          product</seealso> of Set1 and Set2.</p>
        <pre>
1> <input>S1 = sofs:set([1,2]),</input>
<input>S2 = sofs:set([a,b]),</input>
<input>R = sofs:product(S1, S2),</input>
<input>sofs:to_external(R).</input>
[{1,a},{1,b},{2,a},{2,b}]</pre>
        <p><c>product(S1,&nbsp;S2)</c> is equivalent to
          <c>product({S1,&nbsp;S2})</c>.</p>
      </desc>
    </func>
    <func>
      <name>projection(SetFun, Set1) -> Set2</name>
      <fsummary>Return a set of substituted elements.</fsummary>
      <type>
        <v>SetFun = set_fun()</v>
        <v>Set1 = Set2 = set()</v>
      </type>
      <desc>
        <p>Returns the set created by substituting each element of
          Set1 by the result of applying SetFun to the element.</p>
        <p>If SetFun is a number i&nbsp;&gt;=&nbsp;1 and Set1 is a
          relation, then the returned set is 
	  the <seealso marker="#projection">projection</seealso> of Set1 
	  onto coordinate i.</p>
        <pre>
1> <input>S1 = sofs:from_term([{1,a},{2,b},{3,a}]),</input>
<input>S2 = sofs:projection(2, S1),</input>
<input>sofs:to_external(S2).</input>
[a,b]</pre>
      </desc>
    </func>
    <func>
      <name>range(BinRel) -> Set</name>
      <fsummary>Return the range of a binary relation.</fsummary>
      <type>
        <v>BinRel = binary_relation()</v>
        <v>Set = set()</v>
      </type>
      <desc>
        <p>Returns the <seealso marker="#range">range</seealso> of the
          binary relation BinRel.</p>
        <pre>
1> <input>R = sofs:relation([{1,a},{1,b},{2,b},{2,c}]),</input>
<input>S = sofs:range(R),</input>
<input>sofs:to_external(S).</input>
[a,b,c]</pre>
      </desc>
    </func>
    <func>
      <name>relation(Tuples [, Type]) -> Relation</name>
      <fsummary>Create a relation.</fsummary>
      <type>
        <v>N = integer()</v>
        <v>Type = N | type()</v>
        <v>Relation = relation()</v>
        <v>Tuples = [tuple()]</v>
      </type>
      <desc>
        <p>Creates a <seealso marker="#relation">relation</seealso>.
          <c>relation(R,&nbsp;T)</c> is equivalent to
          <c>from_term(R,&nbsp;T)</c>, if T is 
	  a <seealso marker="#type">type</seealso> and the result is a
	  relation. If Type is an integer N, then
          <c>[{atom,&nbsp;...,&nbsp;atom}])</c>, 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. <c>relation([])</c> is
          equivalent to <c>relation([],&nbsp;2)</c>.</p>
      </desc>
    </func>
    <func>
      <name>relation_to_family(BinRel) -> Family</name>
      <fsummary>Create a family from a binary relation.</fsummary>
      <type>
        <v>Family = family()</v>
        <v>BinRel = binary_relation()</v>
      </type>
      <desc>
        <p>Returns the <seealso marker="#family">family</seealso>
          Family such that the index set is equal to 
	  the <seealso marker="#domain">domain</seealso> of the binary
	  relation BinRel, and Family[i] is 
	  the <seealso marker="#image">image</seealso> of the set of i 
	  under BinRel.</p>
        <pre>
1> <input>R = sofs:relation([{b,1},{c,2},{c,3}]),</input>
<input>F = sofs:relation_to_family(R),</input>
<input>sofs:to_external(F).</input>
[{b,[1]},{c,[2,3]}]</pre>
      </desc>
    </func>
    <func>
      <name>relative_product(TupleOfBinRels [, BinRel1]) -> BinRel2</name>
      <fsummary>Return the relative product of a tuple of binary relations
         and a binary relation.</fsummary>
      <type>
        <v>TupleOfBinRels = tuple-of(BinRel)</v>
        <v>BinRel = BinRel1 = BinRel2 = binary_relation()</v>
      </type>
      <desc>
        <p>If TupleOfBinRels is a non-empty tuple
          {R[1],&nbsp;...,&nbsp;R[n]} of binary relations and BinRel1
          is a binary relation, then BinRel2 is 
	  the <seealso marker="#tuple_relative_product">relative 
	  product</seealso> of the ordered set (R[i],&nbsp;...,&nbsp;R[n]) 
	  and BinRel1.</p>
        <p>If BinRel1 is omitted, the relation of equality between the
          elements of 
	  the <seealso marker="#Cartesian_product_tuple">Cartesian 
	  product</seealso> of the ranges of R[i],
          range&nbsp;R[1]&nbsp;&times;&nbsp;...&nbsp;&times;&nbsp;range&nbsp;R[n],
          is used instead (intuitively, nothing is "lost").</p>
        <pre>
1> <input>TR = sofs:relation([{1,a},{1,aa},{2,b}]),</input>
<input>R1 = sofs:relation([{1,u},{2,v},{3,c}]),</input>
<input>R2 = sofs:relative_product({TR, R1}),</input>
<input>sofs:to_external(R2).</input>
[{1,{a,u}},{1,{aa,u}},{2,{b,v}}]</pre>
        <p>Note that <c>relative_product({R1},&nbsp;R2)</c> is
          different from <c>relative_product(R1,&nbsp;R2)</c>; the
          tuple of one element is not identified with the element
          itself.</p>
      </desc>
    </func>
    <func>
      <name>relative_product(BinRel1, BinRel2) -> BinRel3</name>
      <fsummary>Return the relative product of 
        two binary relations.</fsummary>
      <type>
        <v>BinRel1 = BinRel2 = BinRel3 = binary_relation()</v>
      </type>
      <desc>
        <p><marker id="relprod_impl"></marker>Returns
          the <seealso marker="#relative_product">relative 
	  product</seealso> of the binary relations BinRel1 and BinRel2.</p>
      </desc>
    </func>
    <func>
      <name>relative_product1(BinRel1, BinRel2) -> BinRel3</name>
      <fsummary>Return the relative_product of 
        two binary relations.</fsummary>
      <type>
        <v>BinRel1 = BinRel2 = BinRel3 = binary_relation()</v>
      </type>
      <desc>
        <p>Returns the <seealso marker="#relative_product">relative 
	  product</seealso> of 
	  the <seealso marker="#converse">converse</seealso> of the
	  binary relation BinRel1 and the binary relation BinRel2.</p>
        <pre>
1> <input>R1 = sofs:relation([{1,a},{1,aa},{2,b}]),</input>
<input>R2 = sofs:relation([{1,u},{2,v},{3,c}]),</input>
<input>R3 = sofs:relative_product1(R1, R2),</input>
<input>sofs:to_external(R3).</input>
[{a,u},{aa,u},{b,v}]</pre>
        <p><c>relative_product1(R1,&nbsp;R2)</c> is equivalent to
          <c>relative_product(converse(R1),&nbsp;R2)</c>.</p>
      </desc>
    </func>
    <func>
      <name>restriction(BinRel1, Set) -> BinRel2</name>
      <fsummary>Return a restriction of a binary relation.</fsummary>
      <type>
        <v>BinRel1 = BinRel2 = binary_relation()</v>
        <v>Set = set()</v>
      </type>
      <desc>
        <p>Returns the <seealso marker="#restriction">restriction</seealso> of
          the binary relation BinRel1 to Set.</p>
        <pre>
1> <input>R1 = sofs:relation([{1,a},{2,b},{3,c}]),</input>
<input>S = sofs:set([1,2,4]),</input>
<input>R2 = sofs:restriction(R1, S),</input>
<input>sofs:to_external(R2).</input>
[{1,a},{2,b}]</pre>
      </desc>
    </func>
    <func>
      <name>restriction(SetFun, Set1, Set2) -> Set3</name>
      <fsummary>Return a restriction of a set.</fsummary>
      <type>
        <v>SetFun = set_fun()</v>
        <v>Set1 = Set2 = Set3 = set()</v>
      </type>
      <desc>
        <p>Returns a subset of Set1 containing those elements that
          yield an element in Set2 as the result of applying SetFun.</p>
        <pre>
1> <input>S1 = sofs:relation([{1,a},{2,b},{3,c}]),</input>
<input>S2 = sofs:set([b,c,d]),</input>
<input>S3 = sofs:restriction(2, S1, S2),</input>
<input>sofs:to_external(S3).</input>
[{2,b},{3,c}]</pre>
      </desc>
    </func>
    <func>
      <name>set(Terms [, Type]) -> Set</name>
      <fsummary>Create a set of atoms or any type of sets.</fsummary>
      <type>
        <v>Set = set()</v>
        <v>Terms = [term()]</v>
        <v>Type = type()</v>
      </type>
      <desc>
        <p>Creates an <seealso marker="#sets_definition">unordered 
	  set</seealso>. <c>set(L,&nbsp;T)</c> is equivalent to
          <c>from_term(L,&nbsp;T)</c>, if the result is an unordered
          set. If no <seealso marker="#type">type</seealso> is
          explicitly given, <c>[atom]</c> is used as type of the set.</p>
      </desc>
    </func>
    <func>
      <name>specification(Fun, Set1) -> Set2</name>
      <fsummary>Select a subset using a predicate.</fsummary>
      <type>
        <v>Fun = spec_fun()</v>
        <v>Set1 = Set2 = set()</v>
      </type>
      <desc>
        <p>Returns the set containing every element of Set1 for which
          Fun returns <c>true</c>. If Fun is a tuple
          <c>{external,&nbsp;Fun2}</c>, Fun2 is applied to the
          <seealso marker="#external_set">external set</seealso> of
          each element, otherwise Fun is applied to each element.</p>
        <pre>
1> <input>R1 = sofs:relation([{a,1},{b,2}]),</input>
<input>R2 = sofs:relation([{x,1},{x,2},{y,3}]),</input>
<input>S1 = sofs:from_sets([R1,R2]),</input>
<input>S2 = sofs:specification({sofs,is_a_function}, S1),</input>
<input>sofs:to_external(S2).</input>
[[{a,1},{b,2}]]</pre>
      </desc>
    </func>
    <func>
      <name>strict_relation(BinRel1) -> BinRel2</name>
      <fsummary>Return the strict relation corresponding to 
        a given relation.</fsummary>
      <type>
        <v>BinRel1 = BinRel2 = binary_relation()</v>
      </type>
      <desc>
        <p>Returns the <seealso marker="#strict_relation">strict 
          relation</seealso> corresponding to the binary relation BinRel1.</p>
        <pre>
1> <input>R1 = sofs:relation([{1,1},{1,2},{2,1},{2,2}]),</input>
<input>R2 = sofs:strict_relation(R1),</input>
<input>sofs:to_external(R2).</input>
[{1,2},{2,1}]</pre>
      </desc>
    </func>
    <func>
      <name>substitution(SetFun, Set1) -> Set2</name>
      <fsummary>Return a function with a given set as domain.</fsummary>
      <type>
        <v>SetFun = set_fun()</v>
        <v>Set1 = Set2 = set()</v>
      </type>
      <desc>
        <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.</p>
        <pre>
1> <input>L = [{a,1},{b,2}].</input>
[{a,1},{b,2}]
2> <input>sofs:to_external(sofs:projection(1,sofs:relation(L))).</input>
[a,b]
3> <input>sofs:to_external(sofs:substitution(1,sofs:relation(L))).</input>
[{{a,1},a},{{b,2},b}]
4> <input>SetFun = {external, fun({A,_}=E) -> {E,A} end},</input>
<input>sofs:to_external(sofs:projection(SetFun,sofs:relation(L))).</input>
[{{a,1},a},{{b,2},b}]</pre>
        <p>The relation of equality between the elements of {a,b,c}:</p>
        <pre>
1> <input>I = sofs:substitution(fun(A) -> A end, sofs:set([a,b,c])),</input>
<input>sofs:to_external(I).</input>
[{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 <seealso marker="#image">image</seealso>
          of Set under BinRel is returned by this function:</p>
        <pre>
images(SetOfSets, BinRel) ->
   Fun = fun(Set) -> 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 <c>images/2</c>, 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:</p>
        <pre>
images2(SetOfSets, BinRel) ->
   CR = sofs:canonical_relation(SetOfSets),
   R = sofs:relative_product1(CR, BinRel),
   sofs:relation_to_family(R).</pre>
      </desc>
    </func>
    <func>
      <name>symdiff(Set1, Set2) -> Set3</name>
      <fsummary>Return the symmetric difference of two sets.</fsummary>
      <type>
        <v>Set1 = Set2 = Set3 = set()</v>
      </type>
      <desc>
        <p>Returns the <seealso marker="#symmetric_difference">symmetric 
          difference</seealso> (or the Boolean sum) of Set1 and Set2.</p>
        <pre>
1> <input>S1 = sofs:set([1,2,3]),</input>
<input>S2 = sofs:set([2,3,4]),</input>
<input>P = sofs:symdiff(S1, S2),</input>
<input>sofs:to_external(P).</input>
[1,4]</pre>
      </desc>
    </func>
    <func>
      <name>symmetric_partition(Set1, Set2) -> {Set3, Set4, Set5}</name>
      <fsummary>Return a partition of two sets.</fsummary>
      <type>
        <v>Set1 = Set2 = Set3 = Set4 = Set5 = set()</v>
      </type>
      <desc>
        <p>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.</p>
      </desc>
    </func>
    <func>
      <name>to_external(AnySet) -> ExternalSet</name>
      <fsummary>Return the elements of a set.</fsummary>
      <type>
        <v>ExternalSet = external_set()</v>
        <v>AnySet = anyset()</v>
      </type>
      <desc>
        <p>Returns the <seealso marker="#external_set">external 
          set</seealso> of an atomic, ordered or unordered set.</p>
      </desc>
    </func>
    <func>
      <name>to_sets(ASet) -> Sets</name>
      <fsummary>Return a list or a tuple of the elements of set.</fsummary>
      <type>
        <v>ASet = set() | ordset()</v>
        <v>Sets = tuple_of(AnySet) | [AnySet]</v>
      </type>
      <desc>
        <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.</p>
      </desc>
    </func>
    <func>
      <name>type(AnySet) -> Type</name>
      <fsummary>Return the type of a set.</fsummary>
      <type>
        <v>AnySet = anyset()</v>
        <v>Type = type()</v>
      </type>
      <desc>
        <p>Returns the <seealso marker="#type">type</seealso> of an
          atomic, ordered or unordered set.</p>
      </desc>
    </func>
    <func>
      <name>union(SetOfSets) -> Set</name>
      <fsummary>Return the union of a set of sets.</fsummary>
      <type>
        <v>Set = set()</v>
        <v>SetOfSets = set_of_sets()</v>
      </type>
      <desc>
        <p>Returns the <seealso marker="#union_n">union</seealso> of the
          set of sets SetOfSets.</p>
      </desc>
    </func>
    <func>
      <name>union(Set1, Set2) -> Set3</name>
      <fsummary>Return the union of two sets.</fsummary>
      <type>
        <v>Set1 = Set2 = Set3 = set()</v>
      </type>
      <desc>
        <p>Returns the <seealso marker="#union">union</seealso> of
          Set1 and Set2.</p>
      </desc>
    </func>
    <func>
      <name>union_of_family(Family) -> Set</name>
      <fsummary>Return the union of a family.</fsummary>
      <type>
        <v>Family = family()</v>
        <v>Set = set()</v>
      </type>
      <desc>
        <p>Returns the union of 
          the <seealso marker="#family">family</seealso> Family.</p>
        <pre>
1> <input>F = sofs:family([{a,[0,2,4]},{b,[0,1,2]},{c,[2,3]}]),</input>
<input>S = sofs:union_of_family(F),</input>
<input>sofs:to_external(S).</input>
[0,1,2,3,4]</pre>
      </desc>
    </func>
    <func>
      <name>weak_relation(BinRel1) -> BinRel2</name>
      <fsummary>Return the weak relation corresponding to 
        a given relation.</fsummary>
      <type>
        <v>BinRel1 = BinRel2 = binary_relation()</v>
      </type>
      <desc>
        <p>Returns a subset S of the <seealso marker="#weak_relation">weak
          relation</seealso> W
          corresponding to the binary relation BinRel1. Let F be the
          <seealso marker="#field">field</seealso> 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.</p>
        <pre>
1> <input>R1 = sofs:relation([{1,1},{1,2},{3,1}]),</input>
<input>R2 = sofs:weak_relation(R1),</input>
<input>sofs:to_external(R2).</input>
[{1,1},{1,2},{2,2},{3,1},{3,3}]</pre>
      </desc>
    </func>
  </funcs>

  <section>
    <title>See Also</title>
    <p><seealso marker="dict">dict(3)</seealso>, 
      <seealso marker="digraph">digraph(3)</seealso>,
      <seealso marker="orddict">orddict(3)</seealso>, 
      <seealso marker="ordsets">ordsets(3)</seealso>, 
      <seealso marker="sets">sets(3)</seealso></p>
  </section>
</erlref>