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

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<!DOCTYPE erlref SYSTEM "erlref.dtd">

<erlref>
  <header>
    <copyright>
      <year>2001</year><year>2013</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><marker id="binary_relation"></marker>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 <marker id="n_ary_relation"></marker>
      <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 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. <marker id="set_fun"></marker>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>
fun sofs:union/1
fun(S) -> sofs:partition(1, S) end
{external, fun(A) -> A end}
{external, fun({A,_,C}) -> {C,A} end}
{external, fun({_,{_,C}}) -> C end}
{external, fun({_,{_,{_,E}=C}}) -> {E,{E,C}} end}
2</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>
  </description>
  <datatypes>
    <datatype>
      <name name="anyset"></name>
      <desc><p>Any kind of set (also included are the atomic sets).</p></desc>
    </datatype>
    <datatype>
      <name name="binary_relation"></name>
      <desc><p>A <seealso marker="#binary_relation">binary
        relation</seealso>.</p></desc>
    </datatype>
    <datatype>
      <name name="external_set"></name>
      <desc><p>An <seealso marker="#external_set">external
        set</seealso>.</p></desc>
    </datatype>
    <datatype>
      <name name="family"></name>
      <desc><p>A <seealso marker="#family">family</seealso> (of subsets).</p>
      </desc>
    </datatype>
    <datatype>
      <name name="a_function"></name>
      <desc><p>A <seealso marker="#function">function</seealso>.</p></desc>
    </datatype>
    <datatype>
      <name name="ordset"></name>
      <desc><p>An <seealso marker="#sets_definition">ordered
        set</seealso>.</p></desc>
    </datatype>
    <datatype>
      <name name="relation"></name>
      <desc><p>An <seealso marker="#n_ary_relation">n-ary relation</seealso>.
      </p></desc>
    </datatype>
    <datatype>
      <name name="a_set"></name>
      <desc><p>An <seealso marker="#sets_definition">unordered
        set</seealso>.</p></desc>
    </datatype>
    <datatype>
      <name name="set_of_sets"></name>
      <desc><p>An <seealso marker="#sets_definition">unordered
        set</seealso> of unordered sets.</p></desc>
    </datatype>
    <datatype>
      <name name="set_fun"></name>
      <desc><p>A <seealso marker="#set_fun">SetFun</seealso>.</p></desc>
    </datatype>
    <datatype>
      <name name="spec_fun"></name>
    </datatype>
    <datatype>
      <name name="type"></name>
      <desc><p>A <seealso marker="#type">type</seealso>.</p></desc>
    </datatype>
    <datatype>
      <!-- Parameterized opaque types are NYI: -->
      <name><marker id="type-tuple_of">tuple_of(T)</marker></name>
      <desc><p>A tuple where the elements are of type <c>T</c>.</p></desc>
    </datatype>
  </datatypes>
  <funcs>
    <func>
      <name name="a_function" arity="1"/>
      <name name="a_function" arity="2"/>
      <fsummary>Create a function.</fsummary>
      <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 name="canonical_relation" arity="1"/>
      <fsummary>Return the canonical map.</fsummary>
      <desc>
        <p>Returns the binary relation containing the elements
          (E,&nbsp;Set) such that Set belongs to <anno>SetOfSets</anno> 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 name="composite" arity="2"/>
      <fsummary>Return the composite of two functions.</fsummary>
      <desc>
        <p>Returns the <seealso marker="#composite">composite</seealso> of 
	  the functions <anno>Function1</anno> and
	  <anno>Function2</anno>.</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 name="constant_function" arity="2"/>
      <fsummary>Create the function that maps each element of a 
        set onto another set.</fsummary>
      <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 name="converse" arity="1"/>
      <fsummary>Return the converse of a binary relation.</fsummary>
      <desc>
        <p>Returns the <seealso marker="#converse">converse</seealso>
          of the binary relation <anno>BinRel1</anno>.</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 name="difference" arity="2"/>
      <fsummary>Return the difference of two sets.</fsummary>
      <desc>
        <p>Returns the <seealso marker="#difference">difference</seealso> of 
	  the sets <anno>Set1</anno> and <anno>Set2</anno>.</p>
      </desc>
    </func>
    <func>
      <name name="digraph_to_family" arity="1"/>
      <name name="digraph_to_family" arity="2"/>
      <fsummary>Create a family from a directed graph.</fsummary>
      <desc>
        <p>Creates a <seealso marker="#family">family</seealso> from
          the directed graph <anno>Graph</anno>. Each vertex a of
          <anno>Graph</anno> 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 <anno>Type</anno> 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 name="domain" arity="1"/>
      <fsummary>Return the domain of a binary relation.</fsummary>
      <desc>
        <p>Returns the <seealso marker="#domain">domain</seealso> of 
	  the binary relation <anno>BinRel</anno>.</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 name="drestriction" arity="2"/>
      <fsummary>Return a restriction of a binary relation.</fsummary>
      <desc>
        <p>Returns the difference between the binary relation
          <anno>BinRel1</anno>
          and the <seealso marker="#restriction">restriction</seealso>
          of <anno>BinRel1</anno> to <anno>Set</anno>.</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 name="drestriction" arity="3"/>
      <fsummary>Return a restriction of a relation.</fsummary>
      <desc>
        <p>Returns a subset of <anno>Set1</anno> containing those elements
          that do
          not yield an element in <anno>Set2</anno> as the result of applying
          <anno>SetFun</anno>.</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 name="empty_set" arity="0"/>
      <fsummary>Return the untyped empty set.</fsummary>
      <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 name="extension" arity="3"/>
      <fsummary>Extend the domain of a binary relation.</fsummary>
      <desc>
        <p>Returns the <seealso marker="#extension">extension</seealso> of 
	  <anno>BinRel1</anno> such that
          for each element E in <anno>Set</anno> that does not belong to the
          <seealso marker="#domain">domain</seealso> of <anno>BinRel1</anno>,
          <anno>BinRel2</anno> 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 name="family" arity="1"/>
      <name name="family" arity="2"/>
      <fsummary>Create a family of subsets.</fsummary>
      <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 name="family_difference" arity="2"/>
      <fsummary>Return the difference of two families.</fsummary>
      <desc>
        <p>If <anno>Family1</anno> and <anno>Family2</anno>
	  are <seealso marker="#family">families</seealso>, then 
	  <anno>Family3</anno> is the family
          such that the index set is equal to the index set of
          <anno>Family1</anno>, and <anno>Family3</anno>[i] is the
          difference between <anno>Family1</anno>[i]
          and <anno>Family2</anno>[i] if <anno>Family2</anno> maps i,
          <anno>Family1</anno>[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 name="family_domain" arity="1"/>
      <fsummary>Return a family of domains.</fsummary>
      <desc>
        <p>If <anno>Family1</anno> is
          a <seealso marker="#family">family</seealso>
          and <anno>Family1</anno>[i] is a binary relation for every i
          in the index set of <anno>Family1</anno>,
          then <anno>Family2</anno> is the family with the same index
          set as <anno>Family1</anno> such
          that <anno>Family2</anno>[i] is
          the <seealso marker="#domain">domain</seealso> of
          <anno>Family1</anno>[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 name="family_field" arity="1"/>
      <fsummary>Return a family of fields.</fsummary>
      <desc>
        <p>If <anno>Family1</anno> is
          a <seealso marker="#family">family</seealso>
          and <anno>Family1</anno>[i] is a binary relation for every i
          in the index set of <anno>Family1</anno>,
          then <anno>Family2</anno> is the family with the same index
          set as <anno>Family1</anno> such
          that <anno>Family2</anno>[i] is
          the <seealso marker="#field">field</seealso> of
          <anno>Family1</anno>[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 name="family_intersection" arity="1"/>
      <fsummary>Return the intersection of a family
         of sets of sets.</fsummary>
      <desc>
        <p>If <anno>Family1</anno> is
          a <seealso marker="#family">family</seealso>
          and <anno>Family1</anno>[i] is a set of sets for every i in
          the index set of <anno>Family1</anno>,
          then <anno>Family2</anno> is the family with the same index
          set as <anno>Family1</anno> such
          that <anno>Family2</anno>[i] is
          the <seealso marker="#intersection_n">intersection</seealso>
          of <anno>Family1</anno>[i].</p>
        <p>If <anno>Family1</anno>[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 name="family_intersection" arity="2"/>
      <fsummary>Return the intersection of two families.</fsummary>
      <desc>
        <p>If <anno>Family1</anno> and <anno>Family2</anno>
	  are <seealso marker="#family">families</seealso>,
	  then <anno>Family3</anno> is the family such that the index
	  set is the intersection of <anno>Family1</anno>'s and
	  <anno>Family2</anno>'s index sets,
	  and <anno>Family3</anno>[i] is the intersection of
	  <anno>Family1</anno>[i] and <anno>Family2</anno>[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 name="family_projection" arity="2"/>
      <fsummary>Return a family of modified subsets.</fsummary>
      <desc>
        <p>If <anno>Family1</anno> is
          a <seealso marker="#family">family</seealso>
          then <anno>Family2</anno> is the family with the same index
          set as <anno>Family1</anno> such
          that <anno>Family2</anno>[i] is the result of
          calling <anno>SetFun</anno> with <anno>Family1</anno>[i] as
          argument.</p>
        <pre>
1> <input>F1 = sofs:from_term([{a,[[1,2],[2,3]]},{b,[[]]}]),</input>
<input>F2 = sofs:family_projection(fun sofs:union/1, F1),</input>
<input>sofs:to_external(F2).</input>
[{a,[1,2,3]},{b,[]}]</pre>
      </desc>
    </func>
    <func>
      <name name="family_range" arity="1"/>
      <fsummary>Return a family of ranges.</fsummary>
      <desc>
        <p>If <anno>Family1</anno> is
          a <seealso marker="#family">family</seealso>
          and <anno>Family1</anno>[i] is a binary relation for every i
          in the index set of <anno>Family1</anno>,
          then <anno>Family2</anno> is the family with the same index
          set as <anno>Family1</anno> such
          that <anno>Family2</anno>[i] is
          the <seealso marker="#range">range</seealso> of
          <anno>Family1</anno>[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 name="family_specification" arity="2"/>
      <fsummary>Select a subset of a family using a predicate.</fsummary>
      <desc>
        <p>If <anno>Family1</anno> is
          a <seealso marker="#family">family</seealso>,
          then <anno>Family2</anno> is
          the <seealso marker="#restriction">restriction</seealso> of
          <anno>Family1</anno> to those elements i of the index set
          for which <anno>Fun</anno> applied
          to <anno>Family1</anno>[i] returns
          <c>true</c>. If <anno>Fun</anno> is a
          tuple <c>{external,&nbsp;Fun2}</c>, Fun2 is applied to
          the <seealso marker="#external_set">external set</seealso>
          of <anno>Family1</anno>[i], otherwise <anno>Fun</anno> is
          applied to <anno>Family1</anno>[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 name="family_to_digraph" arity="1"/>
      <name name="family_to_digraph" arity="2"/>
      <fsummary>Create a directed graph from a family.</fsummary>
      <desc>
        <p>Creates a directed graph from 
	  the <seealso marker="#family">family</seealso> <anno>Family</anno>.
          For each pair (a,&nbsp;{b[1],&nbsp;...,&nbsp;b[n]})
          of <anno>Family</anno>, 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 <seealso marker="digraph#new/0">
          digraph:new/0</seealso> is used for
          creating the directed graph, otherwise the <anno>GraphType</anno>
          argument is passed on as second argument to
          <seealso marker="digraph#new/1">digraph:new/1</seealso>.</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 name="family_to_relation" arity="1"/>
      <fsummary>Create a binary relation from a family.</fsummary>
      <desc>
        <p>If <anno>Family</anno> is
          a <seealso marker="#family">family</seealso>,
          then <anno>BinRel</anno> is the binary relation containing
          all pairs (i,&nbsp;x) such that i belongs to the index set
          of <anno>Family</anno> and x belongs
          to <anno>Family</anno>[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 name="family_union" arity="1"/>
      <fsummary>Return the union of a family of sets of sets.</fsummary>
      <desc>
        <p>If <anno>Family1</anno> is
          a <seealso marker="#family">family</seealso>
          and <anno>Family1</anno>[i] is a set of sets for each i in
          the index set of <anno>Family1</anno>,
          then <anno>Family2</anno> is the family with the same index
          set as <anno>Family1</anno> such
          that <anno>Family2</anno>[i] is
          the <seealso marker="#union_n">union</seealso> of
          <anno>Family1</anno>[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(fun sofs:union/1,&nbsp;F)</c>.</p>
      </desc>
    </func>
    <func>
      <name name="family_union" arity="2"/>
      <fsummary>Return the union of two families.</fsummary>
      <desc>
        <p>If <anno>Family1</anno> and <anno>Family2</anno>
	  are <seealso marker="#family">families</seealso>,
	  then <anno>Family3</anno> is the family such that the index
	  set is the union of <anno>Family1</anno>'s
	  and <anno>Family2</anno>'s index sets,
	  and <anno>Family3</anno>[i] is the union
	  of <anno>Family1</anno>[i] and <anno>Family2</anno>[i] if
	  both maps i, <anno>Family1</anno>[i]
	  or <anno>Family2</anno>[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 name="field" arity="1"/>
      <fsummary>Return the field of a binary relation.</fsummary>
      <desc>
        <p>Returns the <seealso marker="#field">field</seealso> of the
          binary relation <anno>BinRel</anno>.</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 name="from_external" arity="2"/>
      <fsummary>Create a set.</fsummary>
      <desc>
        <p>Creates a set from the <seealso marker="#external_set">external 
	  set</seealso> <anno>ExternalSet</anno>
          and the <seealso marker="#type">type</seealso> <anno>Type</anno>.
          It is assumed that <anno>Type</anno> is
          a <seealso marker="#valid_type">valid
	  type</seealso> of <anno>ExternalSet</anno>.</p>
      </desc>
    </func>
    <func>
      <name name="from_sets" arity="1" clause_i="1"/>
      <fsummary>Create a set out of a list of sets.</fsummary>
      <desc>
        <p>Returns the <seealso marker="#sets_definition">unordered 
          set</seealso> containing the sets of the list
          <anno>ListOfSets</anno>.</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 name="from_sets" arity="1" clause_i="2"/>
      <fsummary>Create an ordered set out of a tuple of sets.</fsummary>
      <desc>
        <p>Returns the <seealso marker="#sets_definition">ordered 
	  set</seealso> containing the sets of the non-empty tuple 
	  <anno>TupleOfSets</anno>.</p>
      </desc>
    </func>
    <func>
      <name name="from_term" arity="1"/>
      <name name="from_term" arity="2"/>
      <fsummary>Create a set.</fsummary>
      <desc>
        <p><marker id="from_term"></marker>Creates an element 
	  of <seealso marker="#sets_definition">Sets</seealso> by 
	  traversing the term <anno>Term</anno>, 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>
          <anno>Type</anno>
          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 name="image" arity="2"/>
      <fsummary>Return the image of a set under a binary relation.</fsummary>
      <desc>
        <p>Returns the <seealso marker="#image">image</seealso> of the
          set <anno>Set1</anno> under the binary
          relation <anno>BinRel</anno>.</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 name="intersection" arity="1"/>
      <fsummary>Return the intersection of a set of sets.</fsummary>
      <desc>
        <p>Returns 
	  the <seealso marker="#intersection_n">intersection</seealso> of 
	  the set of sets <anno>SetOfSets</anno>.</p>
        <p>Intersecting an empty set of sets exits the process with a
          <c>badarg</c> message.</p>
      </desc>
    </func>
    <func>
      <name name="intersection" arity="2"/>
      <fsummary>Return the intersection of two sets.</fsummary>
      <desc>
        <p>Returns 
	  the <seealso marker="#intersection">intersection</seealso> of 
	  <anno>Set1</anno> and <anno>Set2</anno>.</p>
      </desc>
    </func>
    <func>
      <name name="intersection_of_family" arity="1"/>
      <fsummary>Return the intersection of a family.</fsummary>
      <desc>
        <p>Returns the intersection of 
	  the <seealso marker="#family">family</seealso> <anno>Family</anno>.
        </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 name="inverse" arity="1"/>
      <fsummary>Return the inverse of a function.</fsummary>
      <desc>
        <p>Returns the <seealso marker="#inverse">inverse</seealso>
          of the function <anno>Function1</anno>.</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 name="inverse_image" arity="2"/>
      <fsummary>Return the inverse image of a set under 
        a binary relation.</fsummary>
      <desc>
        <p>Returns the <seealso marker="#inverse_image">inverse
	  image</seealso> of <anno>Set1</anno> under the binary
	  relation <anno>BinRel</anno>.</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 name="is_a_function" arity="1"/>
      <fsummary>Test for a function.</fsummary>
      <desc>
        <p>Returns <c>true</c> if the binary relation <anno>BinRel</anno>
          is a <seealso marker="#function">function</seealso> or the
          untyped empty set, <c>false</c> otherwise.</p>
      </desc>
    </func>
    <func>
      <name name="is_disjoint" arity="2"/>
      <fsummary>Test for disjoint sets.</fsummary>
      <desc>
        <p>Returns <c>true</c> if <anno>Set1</anno>
	  and <anno>Set2</anno>
	  are <seealso marker="#disjoint">disjoint</seealso>, <c>false</c>
          otherwise.</p>
      </desc>
    </func>
    <func>
      <name name="is_empty_set" arity="1"/>
      <fsummary>Test for an empty set.</fsummary>
      <desc>
        <p>Returns <c>true</c> if <anno>AnySet</anno> is an empty
          unordered set, <c>false</c> otherwise.</p>
      </desc>
    </func>
    <func>
      <name name="is_equal" arity="2"/>
      <fsummary>Test two sets for equality.</fsummary>
      <desc>
        <p>Returns <c>true</c> if the <anno>AnySet1</anno>
	  and <anno>AnySet2</anno>
	  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 name="is_set" arity="1"/>
      <fsummary>Test for an unordered set.</fsummary>
      <desc>
        <p>Returns <c>true</c> if <anno>AnySet</anno> is
	  an <seealso marker="#sets_definition">unordered set</seealso>, and
          <c>false</c> if <anno>AnySet</anno> is an ordered set or an
          atomic set.</p>
      </desc>
    </func>
    <func>
      <name name="is_sofs_set" arity="1"/>
      <fsummary>Test for an unordered set.</fsummary>
      <desc>
        <p>Returns <c>true</c> if <anno>Term</anno> 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 name="is_subset" arity="2"/>
      <fsummary>Test two sets for subset.</fsummary>
      <desc>
        <p>Returns <c>true</c> if <anno>Set1</anno> is
	  a <seealso marker="#subset">subset</seealso>
	  of <anno>Set2</anno>, <c>false</c> otherwise.</p>
      </desc>
    </func>
    <func>
      <name name="is_type" arity="1"/>
      <fsummary>Test for a type.</fsummary>
      <desc>
        <p>Returns <c>true</c> if the term <anno>Term</anno> is
	  a <seealso marker="#type">type</seealso>.</p>
      </desc>
    </func>
    <func>
      <name name="join" arity="4"/>
      <fsummary>Return the join of two relations.</fsummary>
      <desc>
        <p>Returns the <seealso marker="#natural_join">natural 
	  join</seealso> of the relations <anno>Relation1</anno>
	  and <anno>Relation2</anno> on coordinates <anno>I</anno> and
	  <anno>J</anno>.</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 name="multiple_relative_product" arity="2"/>
      <fsummary>Return the multiple relative product of a tuple of binary 
        relations and a relation.</fsummary>
      <desc>
        <p>If <anno>TupleOfBinRels</anno> is a non-empty tuple
          {R[1],&nbsp;...,&nbsp;R[n]} of binary relations
          and <anno>BinRel1</anno> is a binary relation,
          then <anno>BinRel2</anno> is
	  the <seealso marker="#multiple_relative_product">multiple relative 
	  product</seealso> of the ordered set
          (R[i],&nbsp;...,&nbsp;R[n]) and <anno>BinRel1</anno>.</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 name="no_elements" arity="1"/>
      <fsummary>Return the number of elements of a set.</fsummary>
      <desc>
        <p>Returns the number of elements of the ordered or unordered
          set <anno>ASet</anno>.</p>
      </desc>
    </func>
    <func>
      <name name="partition" arity="1"/>
      <fsummary>Return the coarsest partition given a set of sets.</fsummary>
      <desc>
        <p>Returns the <seealso marker="#partition">partition</seealso> of 
	  the union of the set of sets <anno>SetOfSets</anno> such that two
	  elements are considered equal if they belong to the same
	  elements of <anno>SetOfSets</anno>.</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 name="partition" arity="2"/>
      <fsummary>Return a partition of a set.</fsummary>
      <desc>
        <p>Returns the <seealso marker="#partition">partition</seealso> of 
          <anno>Set</anno> such that two elements are considered equal
          if the results of applying <anno>SetFun</anno> 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 name="partition" arity="3"/>
      <fsummary>Return a partition of a set.</fsummary>
      <desc>
        <p>Returns a pair of sets that, regarded as constituting a
          set, forms a <seealso marker="#partition">partition</seealso> of 
	  <anno>Set1</anno>. If the
          result of applying <anno>SetFun</anno> to an element
          of <anno>Set1</anno> yields an element in <anno>Set2</anno>,
          the element belongs to <anno>Set3</anno>, otherwise the
          element belongs to <anno>Set4</anno>.</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 name="partition_family" arity="2"/>
      <fsummary>Return a family indexing a partition.</fsummary>
      <desc>
        <p>Returns the <seealso marker="#family">family</seealso>
          <anno>Family</anno> where the indexed set is
	  a <seealso marker="#partition">partition</seealso>
	  of <anno>Set</anno> such that two elements are considered
	  equal if the results of applying <anno>SetFun</anno> are the
	  same value i. This i is the index that <anno>Family</anno>
	  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 name="product" arity="1"/>
      <fsummary>Return the Cartesian product of a tuple of sets.</fsummary>
      <desc>
        <p>Returns the <seealso marker="#Cartesian_product_tuple">Cartesian 
	  product</seealso> of the non-empty tuple of sets
          <anno>TupleOfSets</anno>. If (x[1],&nbsp;...,&nbsp;x[n]) is
          an element of the n-ary relation <anno>Relation</anno>, then
          x[i] is drawn from element i
          of <anno>TupleOfSets</anno>.</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 name="product" arity="2"/>
      <fsummary>Return the Cartesian product of two sets.</fsummary>
      <desc>
        <p>Returns the <seealso marker="#Cartesian_product">Cartesian 
          product</seealso> of <anno>Set1</anno>
          and <anno>Set2</anno>.</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 name="projection" arity="2"/>
      <fsummary>Return a set of substituted elements.</fsummary>
      <desc>
        <p>Returns the set created by substituting each element of
          <anno>Set1</anno> by the result of
          applying <anno>SetFun</anno> to the element.</p>
        <p>If <anno>SetFun</anno> is a number i&nbsp;&gt;=&nbsp;1 and
          <anno>Set1</anno> is a relation, then the returned set is
          the <seealso marker="#projection">projection</seealso> of
          <anno>Set1</anno> 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 name="range" arity="1"/>
      <fsummary>Return the range of a binary relation.</fsummary>
      <desc>
        <p>Returns the <seealso marker="#range">range</seealso> of the
          binary relation <anno>BinRel</anno>.</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 name="relation" arity="1"/>
      <name name="relation" arity="2"/>
      <fsummary>Create a relation.</fsummary>
      <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 <anno>Type</anno> 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
          <anno>Tuples</anno> is
          used if there is such a tuple. <c>relation([])</c> is
          equivalent to <c>relation([],&nbsp;2)</c>.</p>
      </desc>
    </func>
    <func>
      <name name="relation_to_family" arity="1"/>
      <fsummary>Create a family from a binary relation.</fsummary>
      <desc>
        <p>Returns the <seealso marker="#family">family</seealso>
          <anno>Family</anno> such that the index set is equal to
	  the <seealso marker="#domain">domain</seealso> of the binary
	  relation <anno>BinRel</anno>, and <anno>Family</anno>[i] is
	  the <seealso marker="#image">image</seealso> of the set of i 
	  under <anno>BinRel</anno>.</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 name="relative_product" arity="1"/>
      <name name="relative_product" arity="2" clause_i="1"/>
      <fsummary>Return the relative product of a list of binary relations
         and a binary relation.</fsummary>
      <desc>
        <p>If <anno>ListOfBinRels</anno> is a non-empty list
          [R[1],&nbsp;...,&nbsp;R[n]] of binary relations and
          <anno>BinRel1</anno>
          is a binary relation, then <anno>BinRel2</anno> is the <seealso
          marker="#tuple_relative_product">relative product</seealso>
          of the ordered set (R[i],&nbsp;...,&nbsp;R[n]) and
          <anno>BinRel1</anno>.</p>
        <p>If <anno>BinRel1</anno> 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
          list of one element is not identified with the element
          itself.</p>
      </desc>
    </func>
    <func>
      <name name="relative_product" arity="2" clause_i="2"/>
      <fsummary>Return the relative product of 
        two binary relations.</fsummary>
      <desc>
        <p><marker id="relprod_impl"></marker>Returns
          the <seealso marker="#relative_product">relative 
	  product</seealso> of the binary relations <anno>BinRel1</anno>
          and <anno>BinRel2</anno>.</p>
      </desc>
    </func>
    <func>
      <name name="relative_product1" arity="2"/>
      <fsummary>Return the relative_product of 
        two binary relations.</fsummary>
      <desc>
        <p>Returns the <seealso marker="#relative_product">relative 
	  product</seealso> of 
	  the <seealso marker="#converse">converse</seealso> of the
	  binary relation <anno>BinRel1</anno> and the binary
	  relation <anno>BinRel2</anno>.</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 name="restriction" arity="2"/>
      <fsummary>Return a restriction of a binary relation.</fsummary>
      <desc>
        <p>Returns the <seealso marker="#restriction">restriction</seealso> of
          the binary relation <anno>BinRel1</anno>
          to <anno>Set</anno>.</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 name="restriction" arity="3"/>
      <fsummary>Return a restriction of a set.</fsummary>
      <desc>
        <p>Returns a subset of <anno>Set1</anno> containing those
          elements that yield an element in <anno>Set2</anno> as the
          result of applying <anno>SetFun</anno>.</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 name="set" arity="1"/>
      <name name="set" arity="2"/>
      <fsummary>Create a set of atoms or any type of sets.</fsummary>
      <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 name="specification" arity="2"/>
      <fsummary>Select a subset using a predicate.</fsummary>
      <desc>
        <p>Returns the set containing every element
          of <anno>Set1</anno> for which <anno>Fun</anno>
          returns <c>true</c>. If <anno>Fun</anno> is a tuple
          <c>{external,&nbsp;Fun2}</c>, Fun2 is applied to the
          <seealso marker="#external_set">external set</seealso> of
          each element, otherwise <anno>Fun</anno> 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(fun sofs:is_a_function/1, S1),</input>
<input>sofs:to_external(S2).</input>
[[{a,1},{b,2}]]</pre>
      </desc>
    </func>
    <func>
      <name name="strict_relation" arity="1"/>
      <fsummary>Return the strict relation corresponding to 
        a given relation.</fsummary>
      <desc>
        <p>Returns the <seealso marker="#strict_relation">strict 
          relation</seealso> corresponding to the binary
          relation <anno>BinRel1</anno>.</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 name="substitution" arity="2"/>
      <fsummary>Return a function with a given set as domain.</fsummary>
      <desc>
        <p>Returns a function, the domain of which
          is <anno>Set1</anno>. The value of an element of the domain
          is the result of applying <anno>SetFun</anno> 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 name="symdiff" arity="2"/>
      <fsummary>Return the symmetric difference of two sets.</fsummary>
      <desc>
        <p>Returns the <seealso marker="#symmetric_difference">symmetric 
          difference</seealso> (or the Boolean sum)
          of <anno>Set1</anno> and <anno>Set2</anno>.</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 name="symmetric_partition" arity="2"/>
      <fsummary>Return a partition of two sets.</fsummary>
      <desc>
        <p>Returns a triple of sets: <anno>Set3</anno> contains the
          elements of <anno>Set1</anno> that do not belong
          to <anno>Set2</anno>; <anno>Set4</anno> contains the
          elements of <anno>Set1</anno> that belong
          to <anno>Set2</anno>; <anno>Set5</anno> contains the
          elements of <anno>Set2</anno> that do not belong
          to <anno>Set1</anno>.</p>
      </desc>
    </func>
    <func>
      <name name="to_external" arity="1"/>
      <fsummary>Return the elements of a set.</fsummary>
      <desc>
        <p>Returns the <seealso marker="#external_set">external 
          set</seealso> of an atomic, ordered or unordered set.</p>
      </desc>
    </func>
    <func>
      <name name="to_sets" arity="1"/>
      <fsummary>Return a list or a tuple of the elements of set.</fsummary>
      <desc>
        <p>Returns the elements of the ordered set <anno>ASet</anno>
          as a tuple of sets, and the elements of the unordered set
          <anno>ASet</anno> as a sorted list of sets without
          duplicates.</p>
      </desc>
    </func>
    <func>
      <name name="type" arity="1"/>
      <fsummary>Return the type of a set.</fsummary>
      <desc>
        <p>Returns the <seealso marker="#type">type</seealso> of an
          atomic, ordered or unordered set.</p>
      </desc>
    </func>
    <func>
      <name name="union" arity="1"/>
      <fsummary>Return the union of a set of sets.</fsummary>
      <desc>
        <p>Returns the <seealso marker="#union_n">union</seealso> of the
          set of sets <anno>SetOfSets</anno>.</p>
      </desc>
    </func>
    <func>
      <name name="union" arity="2"/>
      <fsummary>Return the union of two sets.</fsummary>
      <desc>
        <p>Returns the <seealso marker="#union">union</seealso> of
          <anno>Set1</anno> and <anno>Set2</anno>.</p>
      </desc>
    </func>
    <func>
      <name name="union_of_family" arity="1"/>
      <fsummary>Return the union of a family.</fsummary>
      <desc>
        <p>Returns the union of 
          the <seealso marker="#family">family</seealso> <anno>Family</anno>.
        </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 name="weak_relation" arity="1"/>
      <fsummary>Return the weak relation corresponding to 
        a given relation.</fsummary>
      <desc>
        <p>Returns a subset S of the <seealso marker="#weak_relation">weak
          relation</seealso> W
          corresponding to the binary relation <anno>BinRel1</anno>.
          Let F be the <seealso marker="#field">field</seealso> of
          <anno>BinRel1</anno>. 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>