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<img alt="Erlang logo" src="../../../../doc/erlang-logo.png"><br><small><a href="users_guide.html">User's Guide</a><br><a href="index.html">Reference Manual</a><br><a href="release_notes.html">Release Notes</a><br><a href="../pdf/stdlib-1.18.3.pdf">PDF</a><br><a href="../../../../doc/index.html">Top</a></small><p><strong>STDLIB</strong><br><strong>Reference Manual</strong><br><small>Version 1.18.3</small></p>
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<li title="default-1"><a href="array.html#default-1">default/1</a></li>
<li title="fix-1"><a href="array.html#fix-1">fix/1</a></li>
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<li title="format_error-1"><a href="beam_lib.html#format_error-1">format_error/1</a></li>
<li title="crypto_key_fun-1"><a href="beam_lib.html#crypto_key_fun-1">crypto_key_fun/1</a></li>
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<li title="copy-1"><a href="binary.html#copy-1">copy/1</a></li>
<li title="copy-2"><a href="binary.html#copy-2">copy/2</a></li>
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<li title="encode_unsigned-1"><a href="binary.html#encode_unsigned-1">encode_unsigned/1</a></li>
<li title="encode_unsigned-2"><a href="binary.html#encode_unsigned-2">encode_unsigned/2</a></li>
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<li title="part-2"><a href="binary.html#part-2">part/2</a></li>
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<li title="referenced_byte_size-1"><a href="binary.html#referenced_byte_size-1">referenced_byte_size/1</a></li>
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<li title="nc-1"><a href="c.html#nc-1">nc/1</a></li>
<li title="nc-2"><a href="c.html#nc-2">nc/2</a></li>
<li title="nl-1"><a href="c.html#nl-1">nl/1</a></li>
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<li title="day_of_the_week-1"><a href="calendar.html#day_of_the_week-1">day_of_the_week/1</a></li>
<li title="day_of_the_week-3"><a href="calendar.html#day_of_the_week-3">day_of_the_week/3</a></li>
<li title="gregorian_days_to_date-1"><a href="calendar.html#gregorian_days_to_date-1">gregorian_days_to_date/1</a></li>
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<li title="iso_week_number-0"><a href="calendar.html#iso_week_number-0">iso_week_number/0</a></li>
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<li title="last_day_of_the_month-2"><a href="calendar.html#last_day_of_the_month-2">last_day_of_the_month/2</a></li>
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<li title="local_time_to_universal_time-1"><a href="calendar.html#local_time_to_universal_time-1">local_time_to_universal_time/1</a></li>
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<!-- refpage --><center><h1>sofs</h1></center>
<h3>MODULE</h3>
<div class="REFBODY">sofs</div>
<h3>MODULE SUMMARY</h3>
<div class="REFBODY">Functions for Manipulating Sets of Sets</div>
<h3>DESCRIPTION</h3>
<div class="REFBODY"><p>
<p>The <span class="code">sofs</span> 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 =
{x in A : S(x)}. Sentences are expressed using
the logical operators "for some" (or "there exists"), "for all",
"and", "or", "not". If the existence of a set containing all the
specified elements is known (as will always be the case in this
module), we write B = {x : S(x)}. </p>
<p>The <strong>unordered set</strong> containing the elements a, b and c
is denoted {a, b, c}. This notation is not to be
confused with tuples. The <strong>ordered pair</strong> of a and b, with
first <strong>coordinate</strong> a and second coordinate b, is denoted
(a, b). An ordered pair is an <strong>ordered set</strong> of two
elements. In this module ordered sets can contain one, two or
more elements, and parentheses are used to enclose the elements.
Unordered sets and ordered sets are orthogonal, again in this
module; there is no unordered set equal to any ordered set.</p>
<p>The set that contains no elements is called the <strong>empty set</strong>.
If two sets A and B contain the same elements, then A
is <a name="equal"></a><strong>equal</strong> to B, denoted
A = B. Two ordered sets are equal if they contain the
same number of elements and have equal elements at each
coordinate. If a set A contains all elements that B contains,
then B is a <a name="subset"></a><strong>subset</strong> of A.
The <a name="union"></a><strong>union</strong> of two sets A and B is
the smallest set that contains all elements of A and all elements of
B. The <a name="intersection"></a><strong>intersection</strong> of two
sets A and B is the set that contains all elements of A that
belong to B.
Two sets are <a name="disjoint"></a><strong>disjoint</strong> if their
intersection is the empty set.
The <a name="difference"></a><strong>difference</strong> of
two sets A and B is the set that contains all elements of A that
do not belong to B.
The <a name="symmetric_difference"></a><strong>symmetric
difference</strong> of
two sets is the set that contains those element that belong to
either of the two sets, but not both.
The <a name="union_n"></a><strong>union</strong> of a collection
of sets is the smallest set that contains all the elements that
belong to at least one set of the collection.
The <a name="intersection_n"></a><strong>intersection</strong> of
a non-empty collection of sets is the set that contains all elements
that belong to every set of the collection.</p>
<p>The <a name="Cartesian_product"></a><strong>Cartesian
product</strong> of
two sets X and Y, denoted X × Y, is the set
{a : a = (x, y) for some x in X and for
some y in Y}.
A <a name="relation"></a><strong>relation</strong> is a subset of
X × Y. Let R be a relation. The fact that
(x, y) belongs to R is written as x R y. Since
relations are sets, the definitions of the last paragraph
(subset, union, and so on) apply to relations as well.
The <a name="domain"></a><strong>domain</strong> of R is the
set {x : x R y for some y in Y}.
The <a name="range"></a><strong>range</strong> of R is the
set {y : x R y for some x in X}.
The <a name="converse"></a><strong>converse</strong> of R is the
set {a : a = (y, x) for some
(x, y) in R}. If A is a subset of X, then
the <a name="image"></a><strong>image</strong> of
A under R is the set {y : x R y for some
x in A}, and if B is a subset of Y, then
the <a name="inverse_image"></a><strong>inverse image</strong> of B is
the set {x : x R y for some y in B}. If R is a
relation from X to Y and S is a relation from Y to Z, then
the <a name="relative_product"></a><strong>relative product</strong> of
R and S is the relation T from X to Z defined so that x T z
if and only if there exists an element y in Y such that
x R y and y S z.
The <a name="restriction"></a><strong>restriction</strong> of R to A is
the set S defined so that x S y if and only if there exists an
element x in A such that x R y. If S is a restriction
of R to A, then R is
an <a name="extension"></a><strong>extension</strong> of S to X.
If X = Y then we call R a relation <strong>in</strong> X.
The <a name="field"></a><strong>field</strong> of a relation R in X
is the union of the domain of R and the range of R.
If R is a relation in X, and
if S is defined so that x S y if x R y and
not x = y, then S is
the <a name="strict_relation"></a><strong>strict</strong> relation
corresponding to
R, and vice versa, if S is a relation in X, and if R is defined
so that x R y if x S y or x = y,
then R is the <a name="weak_relation"></a><strong>weak</strong> relation
corresponding to S. A relation R in X is <strong>reflexive</strong> if
x R x for every element x of X; it is
<strong>symmetric</strong> if x R y implies that
y R x; and it is <strong>transitive</strong> if
x R y and y R z imply that x R z.</p>
<p>A <a name="function"></a><strong>function</strong> F is a relation, a
subset of X × Y, such that the domain of F is
equal to X and such that for every x in X there is a unique
element y in Y with (x, y) in F. The latter condition can
be formulated as follows: if x F y and x F z
then y = z. In this module, it will not be required
that the domain of F be equal to X for a relation to be
considered a function. Instead of writing
(x, y) in F or x F y, we write
F(x) = y when F is a function, and say that F maps x
onto y, or that the value of F at x is y. Since functions are
relations, the definitions of the last paragraph (domain, range,
and so on) apply to functions as well. If the converse of a
function F is a function F', then F' is called
the <a name="inverse"></a><strong>inverse</strong> of F.
The relative product of two functions F1 and F2 is called
the <a name="composite"></a><strong>composite</strong> 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 <strong>family</strong>.
The domain of a family is called the <strong>index set</strong>, and the
range is called the <strong>indexed set</strong>. If x is a family from
I to X, then x[i] denotes the value of the function at index i.
The notation "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 <a name="family"></a><strong>family of subsets</strong> of X. If x
is a family of subsets of X, then the union of the range of x is
called the <strong>union of the family</strong> x. If x is non-empty
(the index set is non-empty),
the <strong>intersection of the family</strong> x is the intersection of
the range of x. In this
module, the only families that will be considered are families
of subsets of some set X; in the following the word "family"
will be used for such families of subsets.</p>
<p>A <a name="partition"></a><strong>partition</strong> of a set X is a
collection S of non-empty subsets of X whose union is X and
whose elements are pairwise disjoint. A relation in a set is an
<strong>equivalence relation</strong> if it is reflexive, symmetric and
transitive. If R is an equivalence relation in X, and x is an
element of X,
the <a name="equivalence_class"></a><strong>equivalence
class</strong> of x with respect to R is the set of all those
elements y of X for which x R y holds. The equivalence
classes constitute a partitioning of X. Conversely, if C is a
partition of X, then the relation that holds for any two
elements of X if they belong to the same equivalence class, is
an equivalence relation induced by the partition C. If R is an
equivalence relation in X, then
the <a name="canonical_map"></a><strong>canonical map</strong> is
the function that maps every element of X onto its equivalence class.
</p>
<p><a name="binary_relation"></a>Relations as defined above
(as sets of ordered pairs) will from now on be referred to as
<strong>binary relations</strong>. We call a set of ordered sets
(x[1], ..., x[n]) an <a name="n_ary_relation"></a>
<strong>(n-ary) relation</strong>, and say that the relation is a subset of
the <a name="Cartesian_product_tuple"></a>Cartesian product
X[1] × ... × X[n] where x[i] is
an element of X[i], 1 <= i <= n.
The <a name="projection"></a><strong>projection</strong> of an n-ary
relation R onto coordinate i is the set {x[i] :
(x[1], ..., x[i], ..., x[n]) in R for some
x[j] in X[j], 1 <= j <= n
and not i = j}. The projections of a binary relation R
onto the first and second coordinates are the domain and the
range of R respectively. The relative product of binary
relations can be generalized to n-ary relations as follows. Let
TR be an ordered set (R[1], ..., R[n]) of binary
relations from X to Y[i] and S a binary relation from
(Y[1] × ... × Y[n]) to Z.
The <a name="tuple_relative_product"></a><strong>relative
product</strong> of
TR and S is the binary relation T from X to Z defined so that
x T z if and only if there exists an element y[i] in
Y[i] for each 1 <= i <= n such that
x R[i] y[i] and
(y[1], ..., y[n]) S z. Now let TR be a an
ordered set (R[1], ..., R[n]) of binary relations from
X[i] to Y[i] and S a subset of
X[1] × ... × X[n].
The <a name="multiple_relative_product"></a><strong>multiple
relative product</strong> of TR and S is defined to be the
set {z : z = ((x[1], ..., x[n]), (y[1],...,y[n]))
for some (x[1], ..., x[n]) in S and for some
(x[i], y[i]) in R[i],
1 <= i <= n}.
The <a name="natural_join"></a><strong>natural join</strong> of
an n-ary relation R
and an m-ary relation S on coordinate i and j is defined to be
the set {z : z = (x[1], ..., x[n],
y[1], ..., y[j-1], y[j+1], ..., y[m])
for some (x[1], ..., x[n]) in R and for some
(y[1], ..., y[m]) in S such that
x[i] = y[j]}.</p>
<p><a name="sets_definition"></a>The sets recognized by this
module will be represented by elements of the relation Sets, defined as
the smallest set such that:</p>
<ul>
<li>for every atom T except '_' and for every term X,
(T, X) belongs to Sets (<strong>atomic sets</strong>);
</li>
<li>(['_'], []) belongs to Sets (the <strong>untyped empty set</strong>);
</li>
<li>for every tuple T = {T[1], ..., T[n]} and
for every tuple X = {X[1], ..., X[n]}, if
(T[i], X[i]) belongs to Sets for every
1 <= i <= n then (T, X) belongs
to Sets (<strong>ordered sets</strong>);
</li>
<li>for every term T, if X is the empty list or a non-empty
sorted list [X[1], ..., X[n]] without duplicates
such that (T, X[i]) belongs to Sets for every
1 <= i <= n, then ([T], X)
belongs to Sets (<strong>typed unordered sets</strong>).</li>
</ul>
<p>An <a name="external_set"></a><strong>external set</strong> is an
element of the range of Sets.
A <a name="type"></a><strong>type</strong>
is an element of the domain of Sets. If S is an element
(T, X) of Sets, then T is
a <a name="valid_type"></a><strong>valid type</strong> of X,
T is the type of S, and X is the external set
of S. <span class="bold_code"><a href="#from_term">from_term/2</a></span> 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>
<ul>
<li>Set(T,Term) = Term, where T is an atom;</li>
<li>Set({T[1], ..., T[n]}, {X[1], ..., X[n]})
= (Set(T[1], X[1]), ..., Set(T[n], X[n]));</li>
<li>Set([T], [X[1], ..., X[n]])
= {Set(T, X[1]), ..., Set(T, X[n])};</li>
<li>Set([T], []) = {}.</li>
</ul>
<p>When there is no risk of confusion, elements of Sets will be
identified with the sets they represent. For instance, if U is
the result of calling <span class="code">union/2</span> 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, <span class="bold_code"><a href="#relprod_impl">relative_product/2</a></span>, 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 (<span class="code">drestriction/3</span>,
<span class="code">family_projection/2</span>, <span class="code">partition/2</span>,
<span class="code">partition_family/2</span>, <span class="code">projection/2</span>,
<span class="code">restriction/3</span>, <span class="code">substitution/2</span>) accept an Erlang
function as a means to modify each element of a given unordered
set. <a name="set_fun"></a>Such a function, called
SetFun in the following, can be
specified as a functional object (fun), a tuple
<span class="code">{external, Fun}</span>, 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 <span class="code">{external, Fun}</span>, 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 <span class="code">{external, fun(X) -> element(I, X) end}</span>,
but is to be preferred since it makes it possible to handle this
case even more efficiently. Examples of SetFuns:</p>
<div class="example"><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></div>
<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: <span class="code">from_external</span>,
<span class="code">is_empty_set</span>, <span class="code">is_set</span>, <span class="code">is_sofs_set</span>,
<span class="code">to_external</span>, <span class="code">type</span>.</p>
<p>The functions of this module exit the process with a
<span class="code">badarg</span>, <span class="code">bad_function</span>, or <span class="code">type_mismatch</span>
message when given badly formed arguments or sets the types of
which are not compatible.</p>
<p>When comparing external sets the operator <span class="code">==/2</span> is used.</p>
</p></div>
<h3>DATA TYPES</h3>
<p>
<span class="bold_code"><a name="type-anyset">anyset()</a> = <span class="bold_code"><a href="#type-ordset">ordset()</a></span> | <span class="bold_code"><a href="#type-a_set">a_set()</a></span></span><br></p>
<div class="REFBODY"><p><p>Any kind of set (also included are the atomic sets).</p></p></div>
<p>
<span class="bold_code"><a name="type-binary_relation">binary_relation()</a> = <span class="bold_code"><a href="#type-relation">relation()</a></span></span><br></p>
<div class="REFBODY"><p><p>A <span class="bold_code"><a href="#binary_relation">binary
relation</a></span>.</p></p></div>
<p>
<span class="bold_code"><a name="type-external_set">external_set()</a> = term()</span><br></p>
<div class="REFBODY"><p><p>An <span class="bold_code"><a href="#external_set">external
set</a></span>.</p></p></div>
<p>
<span class="bold_code"><a name="type-family">family()</a> = <span class="bold_code"><a href="#type-a_function">a_function()</a></span></span><br></p>
<div class="REFBODY"><p><p>A <span class="bold_code"><a href="#family">family</a></span> (of subsets).</p>
</p></div>
<p>
<span class="bold_code"><a name="type-a_function">a_function()</a> = <span class="bold_code"><a href="#type-relation">relation()</a></span></span><br></p>
<div class="REFBODY"><p><p>A <span class="bold_code"><a href="#function">function</a></span>.</p></p></div>
<p>
<span class="bold_code"><a name="type-ordset">ordset()</a></span><br></p>
<div class="REFBODY"><p><p>An <span class="bold_code"><a href="#sets_definition">ordered
set</a></span>.</p></p></div>
<p>
<span class="bold_code"><a name="type-relation">relation()</a> = <span class="bold_code"><a href="#type-a_set">a_set()</a></span></span><br></p>
<div class="REFBODY"><p><p>An <span class="bold_code"><a href="#n_ary_relation">n-ary relation</a></span>.
</p></p></div>
<p>
<span class="bold_code"><a name="type-a_set">a_set()</a></span><br></p>
<div class="REFBODY"><p><p>An <span class="bold_code"><a href="#sets_definition">unordered
set</a></span>.</p></p></div>
<p>
<span class="bold_code"><a name="type-set_of_sets">set_of_sets()</a> = <span class="bold_code"><a href="#type-a_set">a_set()</a></span></span><br></p>
<div class="REFBODY"><p><p>An <span class="bold_code"><a href="#sets_definition">unordered
set</a></span> of unordered sets.</p></p></div>
<p>
<span class="bold_code"><a name="type-set_fun">set_fun()</a> = integer() >= 1<br> | {external, fun((<span class="bold_code"><a href="#type-external_set">external_set()</a></span>) -> <span class="bold_code"><a href="#type-external_set">external_set()</a></span>)}<br> | fun((<span class="bold_code"><a href="#type-anyset">anyset()</a></span>) -> <span class="bold_code"><a href="#type-anyset">anyset()</a></span>)</span><br></p>
<div class="REFBODY"><p><p>A <span class="bold_code"><a href="#set_fun">SetFun</a></span>.</p></p></div>
<p>
<span class="bold_code"><a name="type-spec_fun">spec_fun()</a> = {external, fun((<span class="bold_code"><a href="#type-external_set">external_set()</a></span>) -> boolean())}<br> | fun((<span class="bold_code"><a href="#type-anyset">anyset()</a></span>) -> boolean())</span><br></p>
<p>
<span class="bold_code"><a name="type-type">type()</a> = term()</span><br></p>
<div class="REFBODY"><p><p>A <span class="bold_code"><a href="#type">type</a></span>.</p></p></div>
<p><span class="bold_code"><a name="type-tuple_of">tuple_of(T)</a></span></p>
<div class="REFBODY"><p><p>A tuple where the elements are of type <span class="code">T</span>.</p></p></div>
<h3>EXPORTS</h3>
<p><a name="a_function-1"></a><span class="bold_code">a_function(Tuples) -> Function</span><br><a name="a_function-2"></a><span class="bold_code">a_function(Tuples, Type) -> Function</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">Function = <span class="bold_code"><a href="#type-a_function">a_function()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code">Tuples = [tuple()]</span></div>
<div class="REFTYPES"><span class="bold_code">Type = <span class="bold_code"><a href="#type-type">type()</a></span></span></div>
</div></p>
<div class="REFBODY"><p>
<p>Creates a <span class="bold_code"><a href="#function">function</a></span>.
<span class="code">a_function(F, T)</span> is equivalent to
<span class="code">from_term(F, T)</span>, if the result is a function. If
no <span class="bold_code"><a href="#type">type</a></span> is explicitly
given, <span class="code">[{atom, atom}]</span> is used as type of the
function.</p>
</p></div>
<p><a name="canonical_relation-1"></a><span class="bold_code">canonical_relation(SetOfSets) -> BinRel</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">BinRel = <span class="bold_code"><a href="#type-binary_relation">binary_relation()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code">SetOfSets = <span class="bold_code"><a href="#type-set_of_sets">set_of_sets()</a></span></span></div>
</div></p>
<div class="REFBODY"><p>
<p>Returns the binary relation containing the elements
(E, Set) such that Set belongs to SetOfSets and E
belongs to Set. If SetOfSets is
a <span class="bold_code"><a href="#partition">partition</a></span> of a set X and
R is the equivalence relation in X induced by SetOfSets, then the
returned relation is
the <span class="bold_code"><a href="#canonical_map">canonical map</a></span> from
X onto the equivalence classes with respect to R.</p>
<div class="example"><pre>
1> <span class="bold_code">Ss = sofs:from_term([[a,b],[b,c]]),</span>
<span class="bold_code">CR = sofs:canonical_relation(Ss),</span>
<span class="bold_code">sofs:to_external(CR).</span>
[{a,[a,b]},{b,[a,b]},{b,[b,c]},{c,[b,c]}]</pre></div>
</p></div>
<p><a name="composite-2"></a><span class="bold_code">composite(Function1, Function2) -> Function3</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">Function1 = Function2 = Function3 = <span class="bold_code"><a href="#type-a_function">a_function()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
</div></p>
<div class="REFBODY"><p>
<p>Returns the <span class="bold_code"><a href="#composite">composite</a></span> of
the functions Function1 and
Function2.</p>
<div class="example"><pre>
1> <span class="bold_code">F1 = sofs:a_function([{a,1},{b,2},{c,2}]),</span>
<span class="bold_code">F2 = sofs:a_function([{1,x},{2,y},{3,z}]),</span>
<span class="bold_code">F = sofs:composite(F1, F2),</span>
<span class="bold_code">sofs:to_external(F).</span>
[{a,x},{b,y},{c,y}]</pre></div>
</p></div>
<p><a name="constant_function-2"></a><span class="bold_code">constant_function(Set, AnySet) -> Function</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">AnySet = <span class="bold_code"><a href="#type-anyset">anyset()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code">Function = <span class="bold_code"><a href="#type-a_function">a_function()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code">Set = <span class="bold_code"><a href="#type-a_set">a_set()</a></span></span></div>
</div></p>
<div class="REFBODY"><p>
<p>Creates the <span class="bold_code"><a href="#function">function</a></span>
that maps each element of the set Set onto AnySet.</p>
<div class="example"><pre>
1> <span class="bold_code">S = sofs:set([a,b]),</span>
<span class="bold_code">E = sofs:from_term(1),</span>
<span class="bold_code">R = sofs:constant_function(S, E),</span>
<span class="bold_code">sofs:to_external(R).</span>
[{a,1},{b,1}]</pre></div>
</p></div>
<p><a name="converse-1"></a><span class="bold_code">converse(BinRel1) -> BinRel2</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">BinRel1 = BinRel2 = <span class="bold_code"><a href="#type-binary_relation">binary_relation()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
</div></p>
<div class="REFBODY"><p>
<p>Returns the <span class="bold_code"><a href="#converse">converse</a></span>
of the binary relation BinRel1.</p>
<div class="example"><pre>
1> <span class="bold_code">R1 = sofs:relation([{1,a},{2,b},{3,a}]),</span>
<span class="bold_code">R2 = sofs:converse(R1),</span>
<span class="bold_code">sofs:to_external(R2).</span>
[{a,1},{a,3},{b,2}]</pre></div>
</p></div>
<p><a name="difference-2"></a><span class="bold_code">difference(Set1, Set2) -> Set3</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">Set1 = Set2 = Set3 = <span class="bold_code"><a href="#type-a_set">a_set()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
</div></p>
<div class="REFBODY"><p>
<p>Returns the <span class="bold_code"><a href="#difference">difference</a></span> of
the sets Set1 and Set2.</p>
</p></div>
<p><a name="digraph_to_family-1"></a><span class="bold_code">digraph_to_family(Graph) -> Family</span><br><a name="digraph_to_family-2"></a><span class="bold_code">digraph_to_family(Graph, Type) -> Family</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">Graph = digraph()</span></div>
<div class="REFTYPES"><span class="bold_code">Family = <span class="bold_code"><a href="#type-family">family()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code">Type = <span class="bold_code"><a href="#type-type">type()</a></span></span></div>
</div></p>
<div class="REFBODY"><p>
<p>Creates a <span class="bold_code"><a href="#family">family</a></span> from
the directed graph Graph. Each vertex a of
Graph is
represented by a pair (a, {b[1], ..., b[n]})
where the b[i]'s are the out-neighbours of a. If no type is
explicitly given, [{atom, [atom]}] is used as type of
the family. It is assumed that Type is
a <span class="bold_code"><a href="#valid_type">valid type</a></span> 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
<span class="code">family_to_digraph(digraph_to_family(G))</span>.</p>
</p></div>
<p><a name="domain-1"></a><span class="bold_code">domain(BinRel) -> Set</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">BinRel = <span class="bold_code"><a href="#type-binary_relation">binary_relation()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code">Set = <span class="bold_code"><a href="#type-a_set">a_set()</a></span></span></div>
</div></p>
<div class="REFBODY"><p>
<p>Returns the <span class="bold_code"><a href="#domain">domain</a></span> of
the binary relation BinRel.</p>
<div class="example"><pre>
1> <span class="bold_code">R = sofs:relation([{1,a},{1,b},{2,b},{2,c}]),</span>
<span class="bold_code">S = sofs:domain(R),</span>
<span class="bold_code">sofs:to_external(S).</span>
[1,2]</pre></div>
</p></div>
<p><a name="drestriction-2"></a><span class="bold_code">drestriction(BinRel1, Set) -> BinRel2</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">BinRel1 = BinRel2 = <span class="bold_code"><a href="#type-binary_relation">binary_relation()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
<div class="REFTYPES"><span class="bold_code">Set = <span class="bold_code"><a href="#type-a_set">a_set()</a></span></span></div>
</div></p>
<div class="REFBODY"><p>
<p>Returns the difference between the binary relation
BinRel1
and the <span class="bold_code"><a href="#restriction">restriction</a></span>
of BinRel1 to Set.</p>
<div class="example"><pre>
1> <span class="bold_code">R1 = sofs:relation([{1,a},{2,b},{3,c}]),</span>
<span class="bold_code">S = sofs:set([2,4,6]),</span>
<span class="bold_code">R2 = sofs:drestriction(R1, S),</span>
<span class="bold_code">sofs:to_external(R2).</span>
[{1,a},{3,c}]</pre></div>
<p><span class="code">drestriction(R, S)</span> is equivalent to
<span class="code">difference(R, restriction(R, S))</span>.</p>
</p></div>
<p><a name="drestriction-3"></a><span class="bold_code">drestriction(SetFun, Set1, Set2) -> Set3</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">SetFun = <span class="bold_code"><a href="#type-set_fun">set_fun()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code">Set1 = Set2 = Set3 = <span class="bold_code"><a href="#type-a_set">a_set()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
</div></p>
<div class="REFBODY"><p>
<p>Returns a subset of Set1 containing those elements
that do
not yield an element in Set2 as the result of applying
SetFun.</p>
<div class="example"><pre>
1> <span class="bold_code">SetFun = {external, fun({_A,B,C}) -> {B,C} end},</span>
<span class="bold_code">R1 = sofs:relation([{a,aa,1},{b,bb,2},{c,cc,3}]),</span>
<span class="bold_code">R2 = sofs:relation([{bb,2},{cc,3},{dd,4}]),</span>
<span class="bold_code">R3 = sofs:drestriction(SetFun, R1, R2),</span>
<span class="bold_code">sofs:to_external(R3).</span>
[{a,aa,1}]</pre></div>
<p><span class="code">drestriction(F, S1, S2)</span> is equivalent to
<span class="code">difference(S1, restriction(F, S1, S2))</span>.</p>
</p></div>
<p><a name="empty_set-0"></a><span class="bold_code">empty_set() -> Set</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">Set = <span class="bold_code"><a href="#type-a_set">a_set()</a></span></span></div>
</div></p>
<div class="REFBODY"><p>
<p>Returns the <span class="bold_code"><a href="#sets_definition">untyped empty
set</a></span>. <span class="code">empty_set()</span> is equivalent to
<span class="code">from_term([], ['_'])</span>.</p>
</p></div>
<p><a name="extension-3"></a><span class="bold_code">extension(BinRel1, Set, AnySet) -> BinRel2</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">AnySet = <span class="bold_code"><a href="#type-anyset">anyset()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code">BinRel1 = BinRel2 = <span class="bold_code"><a href="#type-binary_relation">binary_relation()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
<div class="REFTYPES"><span class="bold_code">Set = <span class="bold_code"><a href="#type-a_set">a_set()</a></span></span></div>
</div></p>
<div class="REFBODY"><p>
<p>Returns the <span class="bold_code"><a href="#extension">extension</a></span> of
BinRel1 such that
for each element E in Set that does not belong to the
<span class="bold_code"><a href="#domain">domain</a></span> of BinRel1,
BinRel2 contains the pair (E, AnySet).</p>
<div class="example"><pre>
1> <span class="bold_code">S = sofs:set([b,c]),</span>
<span class="bold_code">A = sofs:empty_set(),</span>
<span class="bold_code">R = sofs:family([{a,[1,2]},{b,[3]}]),</span>
<span class="bold_code">X = sofs:extension(R, S, A),</span>
<span class="bold_code">sofs:to_external(X).</span>
[{a,[1,2]},{b,[3]},{c,[]}]</pre></div>
</p></div>
<p><a name="family-1"></a><span class="bold_code">family(Tuples) -> Family</span><br><a name="family-2"></a><span class="bold_code">family(Tuples, Type) -> Family</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">Family = <span class="bold_code"><a href="#type-family">family()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code">Tuples = [tuple()]</span></div>
<div class="REFTYPES"><span class="bold_code">Type = <span class="bold_code"><a href="#type-type">type()</a></span></span></div>
</div></p>
<div class="REFBODY"><p>
<p>Creates a <span class="bold_code"><a href="#family">family of subsets</a></span>.
<span class="code">family(F, T)</span> is equivalent to
<span class="code">from_term(F, T)</span>, if the result is a family. If
no <span class="bold_code"><a href="#type">type</a></span> is explicitly
given, <span class="code">[{atom, [atom]}]</span> is used as type of the
family.</p>
</p></div>
<p><a name="family_difference-2"></a><span class="bold_code">family_difference(Family1, Family2) -> Family3</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">Family1 = Family2 = Family3 = <span class="bold_code"><a href="#type-family">family()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
</div></p>
<div class="REFBODY"><p>
<p>If Family1 and Family2
are <span class="bold_code"><a href="#family">families</a></span>, 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>
<div class="example"><pre>
1> <span class="bold_code">F1 = sofs:family([{a,[1,2]},{b,[3,4]}]),</span>
<span class="bold_code">F2 = sofs:family([{b,[4,5]},{c,[6,7]}]),</span>
<span class="bold_code">F3 = sofs:family_difference(F1, F2),</span>
<span class="bold_code">sofs:to_external(F3).</span>
[{a,[1,2]},{b,[3]}]</pre></div>
</p></div>
<p><a name="family_domain-1"></a><span class="bold_code">family_domain(Family1) -> Family2</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">Family1 = Family2 = <span class="bold_code"><a href="#type-family">family()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
</div></p>
<div class="REFBODY"><p>
<p>If Family1 is
a <span class="bold_code"><a href="#family">family</a></span>
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 <span class="bold_code"><a href="#domain">domain</a></span> of
Family1[i].</p>
<div class="example"><pre>
1> <span class="bold_code">FR = sofs:from_term([{a,[{1,a},{2,b},{3,c}]},{b,[]},{c,[{4,d},{5,e}]}]),</span>
<span class="bold_code">F = sofs:family_domain(FR),</span>
<span class="bold_code">sofs:to_external(F).</span>
[{a,[1,2,3]},{b,[]},{c,[4,5]}]</pre></div>
</p></div>
<p><a name="family_field-1"></a><span class="bold_code">family_field(Family1) -> Family2</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">Family1 = Family2 = <span class="bold_code"><a href="#type-family">family()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
</div></p>
<div class="REFBODY"><p>
<p>If Family1 is
a <span class="bold_code"><a href="#family">family</a></span>
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 <span class="bold_code"><a href="#field">field</a></span> of
Family1[i].</p>
<div class="example"><pre>
1> <span class="bold_code">FR = sofs:from_term([{a,[{1,a},{2,b},{3,c}]},{b,[]},{c,[{4,d},{5,e}]}]),</span>
<span class="bold_code">F = sofs:family_field(FR),</span>
<span class="bold_code">sofs:to_external(F).</span>
[{a,[1,2,3,a,b,c]},{b,[]},{c,[4,5,d,e]}]</pre></div>
<p><span class="code">family_field(Family1)</span> is equivalent to
<span class="code">family_union(family_domain(Family1), family_range(Family1))</span>.</p>
</p></div>
<p><a name="family_intersection-1"></a><span class="bold_code">family_intersection(Family1) -> Family2</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">Family1 = Family2 = <span class="bold_code"><a href="#type-family">family()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
</div></p>
<div class="REFBODY"><p>
<p>If Family1 is
a <span class="bold_code"><a href="#family">family</a></span>
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 <span class="bold_code"><a href="#intersection_n">intersection</a></span>
of Family1[i].</p>
<p>If Family1[i] is an empty set for some i, then
the process exits with a <span class="code">badarg</span> message.</p>
<div class="example"><pre>
1> <span class="bold_code">F1 = sofs:from_term([{a,[[1,2,3],[2,3,4]]},{b,[[x,y,z],[x,y]]}]),</span>
<span class="bold_code">F2 = sofs:family_intersection(F1),</span>
<span class="bold_code">sofs:to_external(F2).</span>
[{a,[2,3]},{b,[x,y]}]</pre></div>
</p></div>
<p><a name="family_intersection-2"></a><span class="bold_code">family_intersection(Family1, Family2) -> Family3</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">Family1 = Family2 = Family3 = <span class="bold_code"><a href="#type-family">family()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
</div></p>
<div class="REFBODY"><p>
<p>If Family1 and Family2
are <span class="bold_code"><a href="#family">families</a></span>,
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>
<div class="example"><pre>
1> <span class="bold_code">F1 = sofs:family([{a,[1,2]},{b,[3,4]},{c,[5,6]}]),</span>
<span class="bold_code">F2 = sofs:family([{b,[4,5]},{c,[7,8]},{d,[9,10]}]),</span>
<span class="bold_code">F3 = sofs:family_intersection(F1, F2),</span>
<span class="bold_code">sofs:to_external(F3).</span>
[{b,[4]},{c,[]}]</pre></div>
</p></div>
<p><a name="family_projection-2"></a><span class="bold_code">family_projection(SetFun, Family1) -> Family2</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">SetFun = <span class="bold_code"><a href="#type-set_fun">set_fun()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code">Family1 = Family2 = <span class="bold_code"><a href="#type-family">family()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
</div></p>
<div class="REFBODY"><p>
<p>If Family1 is
a <span class="bold_code"><a href="#family">family</a></span>
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>
<div class="example"><pre>
1> <span class="bold_code">F1 = sofs:from_term([{a,[[1,2],[2,3]]},{b,[[]]}]),</span>
<span class="bold_code">F2 = sofs:family_projection(fun sofs:union/1, F1),</span>
<span class="bold_code">sofs:to_external(F2).</span>
[{a,[1,2,3]},{b,[]}]</pre></div>
</p></div>
<p><a name="family_range-1"></a><span class="bold_code">family_range(Family1) -> Family2</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">Family1 = Family2 = <span class="bold_code"><a href="#type-family">family()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
</div></p>
<div class="REFBODY"><p>
<p>If Family1 is
a <span class="bold_code"><a href="#family">family</a></span>
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 <span class="bold_code"><a href="#range">range</a></span> of
Family1[i].</p>
<div class="example"><pre>
1> <span class="bold_code">FR = sofs:from_term([{a,[{1,a},{2,b},{3,c}]},{b,[]},{c,[{4,d},{5,e}]}]),</span>
<span class="bold_code">F = sofs:family_range(FR),</span>
<span class="bold_code">sofs:to_external(F).</span>
[{a,[a,b,c]},{b,[]},{c,[d,e]}]</pre></div>
</p></div>
<p><a name="family_specification-2"></a><span class="bold_code">family_specification(Fun, Family1) -> Family2</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">Fun = <span class="bold_code"><a href="#type-spec_fun">spec_fun()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code">Family1 = Family2 = <span class="bold_code"><a href="#type-family">family()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
</div></p>
<div class="REFBODY"><p>
<p>If Family1 is
a <span class="bold_code"><a href="#family">family</a></span>,
then Family2 is
the <span class="bold_code"><a href="#restriction">restriction</a></span> of
Family1 to those elements i of the index set
for which Fun applied
to Family1[i] returns
<span class="code">true</span>. If Fun is a
tuple <span class="code">{external, Fun2}</span>, Fun2 is applied to
the <span class="bold_code"><a href="#external_set">external set</a></span>
of Family1[i], otherwise Fun is
applied to Family1[i].</p>
<div class="example"><pre>
1> <span class="bold_code">F1 = sofs:family([{a,[1,2,3]},{b,[1,2]},{c,[1]}]),</span>
<span class="bold_code">SpecFun = fun(S) -> sofs:no_elements(S) =:= 2 end,</span>
<span class="bold_code">F2 = sofs:family_specification(SpecFun, F1),</span>
<span class="bold_code">sofs:to_external(F2).</span>
[{b,[1,2]}]</pre></div>
</p></div>
<p><a name="family_to_digraph-1"></a><span class="bold_code">family_to_digraph(Family) -> Graph</span><br><a name="family_to_digraph-2"></a><span class="bold_code">family_to_digraph(Family, GraphType) -> Graph</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">Graph = digraph()</span></div>
<div class="REFTYPES"><span class="bold_code">Family = <span class="bold_code"><a href="#type-family">family()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code">GraphType = [<span class="bold_code"><a href="digraph.html#type-d_type">digraph:d_type()</a></span>]</span></div>
</div></p>
<div class="REFBODY"><p>
<p>Creates a directed graph from
the <span class="bold_code"><a href="#family">family</a></span> Family.
For each pair (a, {b[1], ..., b[n]})
of Family, the vertex
a as well the edges (a, b[i]) for
1 <= i <= n are added to a newly
created directed graph.</p>
<p>If no graph type is given <span class="bold_code"><a href="digraph.html#new-0">
digraph:new/0</a></span> is used for
creating the directed graph, otherwise the GraphType
argument is passed on as second argument to
<span class="bold_code"><a href="digraph.html#new-1">digraph:new/1</a></span>.</p>
<p>It F is a family, it holds that F is a subset of
<span class="code">digraph_to_family(family_to_digraph(F), type(F))</span>.
Equality holds if <span class="code">union_of_family(F)</span> is a subset of
<span class="code">domain(F)</span>.</p>
<p>Creating a cycle in an acyclic graph exits the process with
a <span class="code">cyclic</span> message.</p>
</p></div>
<p><a name="family_to_relation-1"></a><span class="bold_code">family_to_relation(Family) -> BinRel</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">Family = <span class="bold_code"><a href="#type-family">family()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code">BinRel = <span class="bold_code"><a href="#type-binary_relation">binary_relation()</a></span></span></div>
</div></p>
<div class="REFBODY"><p>
<p>If Family is
a <span class="bold_code"><a href="#family">family</a></span>,
then BinRel is the binary relation containing
all pairs (i, x) such that i belongs to the index set
of Family and x belongs
to Family[i].</p>
<div class="example"><pre>
1> <span class="bold_code">F = sofs:family([{a,[]}, {b,[1]}, {c,[2,3]}]),</span>
<span class="bold_code">R = sofs:family_to_relation(F),</span>
<span class="bold_code">sofs:to_external(R).</span>
[{b,1},{c,2},{c,3}]</pre></div>
</p></div>
<p><a name="family_union-1"></a><span class="bold_code">family_union(Family1) -> Family2</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">Family1 = Family2 = <span class="bold_code"><a href="#type-family">family()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
</div></p>
<div class="REFBODY"><p>
<p>If Family1 is
a <span class="bold_code"><a href="#family">family</a></span>
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 <span class="bold_code"><a href="#union_n">union</a></span> of
Family1[i].</p>
<div class="example"><pre>
1> <span class="bold_code">F1 = sofs:from_term([{a,[[1,2],[2,3]]},{b,[[]]}]),</span>
<span class="bold_code">F2 = sofs:family_union(F1),</span>
<span class="bold_code">sofs:to_external(F2).</span>
[{a,[1,2,3]},{b,[]}]</pre></div>
<p><span class="code">family_union(F)</span> is equivalent to
<span class="code">family_projection(fun sofs:union/1, F)</span>.</p>
</p></div>
<p><a name="family_union-2"></a><span class="bold_code">family_union(Family1, Family2) -> Family3</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">Family1 = Family2 = Family3 = <span class="bold_code"><a href="#type-family">family()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
</div></p>
<div class="REFBODY"><p>
<p>If Family1 and Family2
are <span class="bold_code"><a href="#family">families</a></span>,
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>
<div class="example"><pre>
1> <span class="bold_code">F1 = sofs:family([{a,[1,2]},{b,[3,4]},{c,[5,6]}]),</span>
<span class="bold_code">F2 = sofs:family([{b,[4,5]},{c,[7,8]},{d,[9,10]}]),</span>
<span class="bold_code">F3 = sofs:family_union(F1, F2),</span>
<span class="bold_code">sofs:to_external(F3).</span>
[{a,[1,2]},{b,[3,4,5]},{c,[5,6,7,8]},{d,[9,10]}]</pre></div>
</p></div>
<p><a name="field-1"></a><span class="bold_code">field(BinRel) -> Set</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">BinRel = <span class="bold_code"><a href="#type-binary_relation">binary_relation()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code">Set = <span class="bold_code"><a href="#type-a_set">a_set()</a></span></span></div>
</div></p>
<div class="REFBODY"><p>
<p>Returns the <span class="bold_code"><a href="#field">field</a></span> of the
binary relation BinRel.</p>
<div class="example"><pre>
1> <span class="bold_code">R = sofs:relation([{1,a},{1,b},{2,b},{2,c}]),</span>
<span class="bold_code">S = sofs:field(R),</span>
<span class="bold_code">sofs:to_external(S).</span>
[1,2,a,b,c]</pre></div>
<p><span class="code">field(R)</span> is equivalent
to <span class="code">union(domain(R), range(R))</span>.</p>
</p></div>
<p><a name="from_external-2"></a><span class="bold_code">from_external(ExternalSet, Type) -> AnySet</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">ExternalSet = <span class="bold_code"><a href="#type-external_set">external_set()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code">AnySet = <span class="bold_code"><a href="#type-anyset">anyset()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code">Type = <span class="bold_code"><a href="#type-type">type()</a></span></span></div>
</div></p>
<div class="REFBODY"><p>
<p>Creates a set from the <span class="bold_code"><a href="#external_set">external
set</a></span> ExternalSet
and the <span class="bold_code"><a href="#type">type</a></span> Type.
It is assumed that Type is
a <span class="bold_code"><a href="#valid_type">valid
type</a></span> of ExternalSet.</p>
</p></div>
<p><a name="from_sets-1"></a><span class="bold_code">from_sets(ListOfSets) -> Set</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">Set = <span class="bold_code"><a href="#type-a_set">a_set()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code">ListOfSets = [<span class="bold_code"><a href="#type-anyset">anyset()</a></span>]</span></div>
</div></p>
<div class="REFBODY"><p>
<p>Returns the <span class="bold_code"><a href="#sets_definition">unordered
set</a></span> containing the sets of the list
ListOfSets.</p>
<div class="example"><pre>
1> <span class="bold_code">S1 = sofs:relation([{a,1},{b,2}]),</span>
<span class="bold_code">S2 = sofs:relation([{x,3},{y,4}]),</span>
<span class="bold_code">S = sofs:from_sets([S1,S2]),</span>
<span class="bold_code">sofs:to_external(S).</span>
[[{a,1},{b,2}],[{x,3},{y,4}]]</pre></div>
</p></div>
<p><span class="bold_code">from_sets(TupleOfSets) -> Ordset</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">Ordset = <span class="bold_code"><a href="#type-ordset">ordset()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code">TupleOfSets = <span class="bold_code"><a href="#type-tuple_of">tuple_of</a></span>(<span class="bold_code"><a href="#type-anyset">anyset()</a></span>)</span></div>
</div></p>
<div class="REFBODY"><p>
<p>Returns the <span class="bold_code"><a href="#sets_definition">ordered
set</a></span> containing the sets of the non-empty tuple
TupleOfSets.</p>
</p></div>
<p><a name="from_term-1"></a><span class="bold_code">from_term(Term) -> AnySet</span><br><a name="from_term-2"></a><span class="bold_code">from_term(Term, Type) -> AnySet</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">AnySet = <span class="bold_code"><a href="#type-anyset">anyset()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code">Term = term()</span></div>
<div class="REFTYPES"><span class="bold_code">Type = <span class="bold_code"><a href="#type-type">type()</a></span></span></div>
</div></p>
<div class="REFBODY"><p>
<p><a name="from_term"></a>Creates an element
of <span class="bold_code"><a href="#sets_definition">Sets</a></span> by
traversing the term Term, sorting lists,
removing duplicates and
deriving or verifying a <span class="bold_code"><a href="#valid_type">valid
type</a></span> for the so obtained external set. An
explicitly given <span class="bold_code"><a href="#type">type</a></span>
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>
<div class="example"><pre>
1> <span class="bold_code">S = sofs:from_term([{{"foo"},[1,1]},{"foo",[2,2]}],
[{atom,[atom]}]),</span>
<span class="bold_code">sofs:to_external(S).</span>
[{{"foo"},[1]},{"foo",[2]}]</pre></div>
<p><span class="code">from_term</span> can be used for creating atomic or ordered
sets. The only purpose of such a set is that of later
building unordered sets since all functions in this module
that <strong>do</strong> anything operate on unordered sets.
Creating unordered sets from a collection of ordered sets
may be the way to go if the ordered sets are big and one
does not want to waste heap by rebuilding the elements of
the unordered set. An example showing that a set can be
built "layer by layer":</p>
<div class="example"><pre>
1> <span class="bold_code">A = sofs:from_term(a),</span>
<span class="bold_code">S = sofs:set([1,2,3]),</span>
<span class="bold_code">P1 = sofs:from_sets({A,S}),</span>
<span class="bold_code">P2 = sofs:from_term({b,[6,5,4]}),</span>
<span class="bold_code">Ss = sofs:from_sets([P1,P2]),</span>
<span class="bold_code">sofs:to_external(Ss).</span>
[{a,[1,2,3]},{b,[4,5,6]}]</pre></div>
<p>Other functions that create sets are <span class="code">from_external/2</span>
and <span class="code">from_sets/1</span>. Special cases of <span class="code">from_term/2</span>
are <span class="code">a_function/1,2</span>, <span class="code">empty_set/0</span>,
<span class="code">family/1,2</span>, <span class="code">relation/1,2</span>, and <span class="code">set/1,2</span>.</p>
</p></div>
<p><a name="image-2"></a><span class="bold_code">image(BinRel, Set1) -> Set2</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">BinRel = <span class="bold_code"><a href="#type-binary_relation">binary_relation()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code">Set1 = Set2 = <span class="bold_code"><a href="#type-a_set">a_set()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
</div></p>
<div class="REFBODY"><p>
<p>Returns the <span class="bold_code"><a href="#image">image</a></span> of the
set Set1 under the binary
relation BinRel.</p>
<div class="example"><pre>
1> <span class="bold_code">R = sofs:relation([{1,a},{2,b},{2,c},{3,d}]),</span>
<span class="bold_code">S1 = sofs:set([1,2]),</span>
<span class="bold_code">S2 = sofs:image(R, S1),</span>
<span class="bold_code">sofs:to_external(S2).</span>
[a,b,c]</pre></div>
</p></div>
<p><a name="intersection-1"></a><span class="bold_code">intersection(SetOfSets) -> Set</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">Set = <span class="bold_code"><a href="#type-a_set">a_set()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code">SetOfSets = <span class="bold_code"><a href="#type-set_of_sets">set_of_sets()</a></span></span></div>
</div></p>
<div class="REFBODY"><p>
<p>Returns
the <span class="bold_code"><a href="#intersection_n">intersection</a></span> of
the set of sets SetOfSets.</p>
<p>Intersecting an empty set of sets exits the process with a
<span class="code">badarg</span> message.</p>
</p></div>
<p><a name="intersection-2"></a><span class="bold_code">intersection(Set1, Set2) -> Set3</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">Set1 = Set2 = Set3 = <span class="bold_code"><a href="#type-a_set">a_set()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
</div></p>
<div class="REFBODY"><p>
<p>Returns
the <span class="bold_code"><a href="#intersection">intersection</a></span> of
Set1 and Set2.</p>
</p></div>
<p><a name="intersection_of_family-1"></a><span class="bold_code">intersection_of_family(Family) -> Set</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">Family = <span class="bold_code"><a href="#type-family">family()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code">Set = <span class="bold_code"><a href="#type-a_set">a_set()</a></span></span></div>
</div></p>
<div class="REFBODY"><p>
<p>Returns the intersection of
the <span class="bold_code"><a href="#family">family</a></span> Family.
</p>
<p>Intersecting an empty family exits the process with a
<span class="code">badarg</span> message.</p>
<div class="example"><pre>
1> <span class="bold_code">F = sofs:family([{a,[0,2,4]},{b,[0,1,2]},{c,[2,3]}]),</span>
<span class="bold_code">S = sofs:intersection_of_family(F),</span>
<span class="bold_code">sofs:to_external(S).</span>
[2]</pre></div>
</p></div>
<p><a name="inverse-1"></a><span class="bold_code">inverse(Function1) -> Function2</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">Function1 = Function2 = <span class="bold_code"><a href="#type-a_function">a_function()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
</div></p>
<div class="REFBODY"><p>
<p>Returns the <span class="bold_code"><a href="#inverse">inverse</a></span>
of the function Function1.</p>
<div class="example"><pre>
1> <span class="bold_code">R1 = sofs:relation([{1,a},{2,b},{3,c}]),</span>
<span class="bold_code">R2 = sofs:inverse(R1),</span>
<span class="bold_code">sofs:to_external(R2).</span>
[{a,1},{b,2},{c,3}]</pre></div>
</p></div>
<p><a name="inverse_image-2"></a><span class="bold_code">inverse_image(BinRel, Set1) -> Set2</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">BinRel = <span class="bold_code"><a href="#type-binary_relation">binary_relation()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code">Set1 = Set2 = <span class="bold_code"><a href="#type-a_set">a_set()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
</div></p>
<div class="REFBODY"><p>
<p>Returns the <span class="bold_code"><a href="#inverse_image">inverse
image</a></span> of Set1 under the binary
relation BinRel.</p>
<div class="example"><pre>
1> <span class="bold_code">R = sofs:relation([{1,a},{2,b},{2,c},{3,d}]),</span>
<span class="bold_code">S1 = sofs:set([c,d,e]),</span>
<span class="bold_code">S2 = sofs:inverse_image(R, S1),</span>
<span class="bold_code">sofs:to_external(S2).</span>
[2,3]</pre></div>
</p></div>
<p><a name="is_a_function-1"></a><span class="bold_code">is_a_function(BinRel) -> Bool</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">Bool = boolean()</span></div>
<div class="REFTYPES"><span class="bold_code">BinRel = <span class="bold_code"><a href="#type-binary_relation">binary_relation()</a></span></span></div>
</div></p>
<div class="REFBODY"><p>
<p>Returns <span class="code">true</span> if the binary relation BinRel
is a <span class="bold_code"><a href="#function">function</a></span> or the
untyped empty set, <span class="code">false</span> otherwise.</p>
</p></div>
<p><a name="is_disjoint-2"></a><span class="bold_code">is_disjoint(Set1, Set2) -> Bool</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">Bool = boolean()</span></div>
<div class="REFTYPES"><span class="bold_code">Set1 = Set2 = <span class="bold_code"><a href="#type-a_set">a_set()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
</div></p>
<div class="REFBODY"><p>
<p>Returns <span class="code">true</span> if Set1
and Set2
are <span class="bold_code"><a href="#disjoint">disjoint</a></span>, <span class="code">false</span>
otherwise.</p>
</p></div>
<p><a name="is_empty_set-1"></a><span class="bold_code">is_empty_set(AnySet) -> Bool</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">AnySet = <span class="bold_code"><a href="#type-anyset">anyset()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code">Bool = boolean()</span></div>
</div></p>
<div class="REFBODY"><p>
<p>Returns <span class="code">true</span> if AnySet is an empty
unordered set, <span class="code">false</span> otherwise.</p>
</p></div>
<p><a name="is_equal-2"></a><span class="bold_code">is_equal(AnySet1, AnySet2) -> Bool</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">AnySet1 = AnySet2 = <span class="bold_code"><a href="#type-anyset">anyset()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
<div class="REFTYPES"><span class="bold_code">Bool = boolean()</span></div>
</div></p>
<div class="REFBODY"><p>
<p>Returns <span class="code">true</span> if the AnySet1
and AnySet2
are <span class="bold_code"><a href="#equal">equal</a></span>, <span class="code">false</span>
otherwise. This example shows that <span class="code">==/2</span> is used when
comparing sets for equality:</p>
<div class="example"><pre>
1> <span class="bold_code">S1 = sofs:set([1.0]),</span>
<span class="bold_code">S2 = sofs:set([1]),</span>
<span class="bold_code">sofs:is_equal(S1, S2).</span>
true</pre></div>
</p></div>
<p><a name="is_set-1"></a><span class="bold_code">is_set(AnySet) -> Bool</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">AnySet = <span class="bold_code"><a href="#type-anyset">anyset()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code">Bool = boolean()</span></div>
</div></p>
<div class="REFBODY"><p>
<p>Returns <span class="code">true</span> if AnySet is
an <span class="bold_code"><a href="#sets_definition">unordered set</a></span>, and
<span class="code">false</span> if AnySet is an ordered set or an
atomic set.</p>
</p></div>
<p><a name="is_sofs_set-1"></a><span class="bold_code">is_sofs_set(Term) -> Bool</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">Bool = boolean()</span></div>
<div class="REFTYPES"><span class="bold_code">Term = term()</span></div>
</div></p>
<div class="REFBODY"><p>
<p>Returns <span class="code">true</span> if Term is
an <span class="bold_code"><a href="#sets_definition">unordered set</a></span>, an
ordered set or an atomic set, <span class="code">false</span> otherwise.</p>
</p></div>
<p><a name="is_subset-2"></a><span class="bold_code">is_subset(Set1, Set2) -> Bool</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">Bool = boolean()</span></div>
<div class="REFTYPES"><span class="bold_code">Set1 = Set2 = <span class="bold_code"><a href="#type-a_set">a_set()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
</div></p>
<div class="REFBODY"><p>
<p>Returns <span class="code">true</span> if Set1 is
a <span class="bold_code"><a href="#subset">subset</a></span>
of Set2, <span class="code">false</span> otherwise.</p>
</p></div>
<p><a name="is_type-1"></a><span class="bold_code">is_type(Term) -> Bool</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">Bool = boolean()</span></div>
<div class="REFTYPES"><span class="bold_code">Term = term()</span></div>
</div></p>
<div class="REFBODY"><p>
<p>Returns <span class="code">true</span> if the term Term is
a <span class="bold_code"><a href="#type">type</a></span>.</p>
</p></div>
<p><a name="join-4"></a><span class="bold_code">join(Relation1, I, Relation2, J) -> Relation3</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">Relation1 = Relation2 = Relation3 = <span class="bold_code"><a href="#type-relation">relation()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
<div class="REFTYPES"><span class="bold_code">I = J = integer() >= 1</span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
</div></p>
<div class="REFBODY"><p>
<p>Returns the <span class="bold_code"><a href="#natural_join">natural
join</a></span> of the relations Relation1
and Relation2 on coordinates I and
J.</p>
<div class="example"><pre>
1> <span class="bold_code">R1 = sofs:relation([{a,x,1},{b,y,2}]),</span>
<span class="bold_code">R2 = sofs:relation([{1,f,g},{1,h,i},{2,3,4}]),</span>
<span class="bold_code">J = sofs:join(R1, 3, R2, 1),</span>
<span class="bold_code">sofs:to_external(J).</span>
[{a,x,1,f,g},{a,x,1,h,i},{b,y,2,3,4}]</pre></div>
</p></div>
<p><a name="multiple_relative_product-2"></a><span class="bold_code">multiple_relative_product(TupleOfBinRels, BinRel1) -> BinRel2</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">TupleOfBinRels = <span class="bold_code"><a href="#type-tuple_of">tuple_of</a></span>(BinRel)</span></div>
<div class="REFTYPES"><span class="bold_code">BinRel = BinRel1 = BinRel2 = <span class="bold_code"><a href="#type-binary_relation">binary_relation()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
</div></p>
<div class="REFBODY"><p>
<p>If TupleOfBinRels is a non-empty tuple
{R[1], ..., R[n]} of binary relations
and BinRel1 is a binary relation,
then BinRel2 is
the <span class="bold_code"><a href="#multiple_relative_product">multiple relative
product</a></span> of the ordered set
(R[i], ..., R[n]) and BinRel1.</p>
<div class="example"><pre>
1> <span class="bold_code">Ri = sofs:relation([{a,1},{b,2},{c,3}]),</span>
<span class="bold_code">R = sofs:relation([{a,b},{b,c},{c,a}]),</span>
<span class="bold_code">MP = sofs:multiple_relative_product({Ri, Ri}, R),</span>
<span class="bold_code">sofs:to_external(sofs:range(MP)).</span>
[{1,2},{2,3},{3,1}]</pre></div>
</p></div>
<p><a name="no_elements-1"></a><span class="bold_code">no_elements(ASet) -> NoElements</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">ASet = <span class="bold_code"><a href="#type-a_set">a_set()</a></span> | <span class="bold_code"><a href="#type-ordset">ordset()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code">NoElements = integer() >= 0</span></div>
</div></p>
<div class="REFBODY"><p>
<p>Returns the number of elements of the ordered or unordered
set ASet.</p>
</p></div>
<p><a name="partition-1"></a><span class="bold_code">partition(SetOfSets) -> Partition</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">SetOfSets = <span class="bold_code"><a href="#type-set_of_sets">set_of_sets()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code">Partition = <span class="bold_code"><a href="#type-a_set">a_set()</a></span></span></div>
</div></p>
<div class="REFBODY"><p>
<p>Returns the <span class="bold_code"><a href="#partition">partition</a></span> 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>
<div class="example"><pre>
1> <span class="bold_code">Sets1 = sofs:from_term([[a,b,c],[d,e,f],[g,h,i]]),</span>
<span class="bold_code">Sets2 = sofs:from_term([[b,c,d],[e,f,g],[h,i,j]]),</span>
<span class="bold_code">P = sofs:partition(sofs:union(Sets1, Sets2)),</span>
<span class="bold_code">sofs:to_external(P).</span>
[[a],[b,c],[d],[e,f],[g],[h,i],[j]]</pre></div>
</p></div>
<p><a name="partition-2"></a><span class="bold_code">partition(SetFun, Set) -> Partition</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">SetFun = <span class="bold_code"><a href="#type-set_fun">set_fun()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code">Partition = Set = <span class="bold_code"><a href="#type-a_set">a_set()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
</div></p>
<div class="REFBODY"><p>
<p>Returns the <span class="bold_code"><a href="#partition">partition</a></span> of
Set such that two elements are considered equal
if the results of applying SetFun are equal.</p>
<div class="example"><pre>
1> <span class="bold_code">Ss = sofs:from_term([[a],[b],[c,d],[e,f]]),</span>
<span class="bold_code">SetFun = fun(S) -> sofs:from_term(sofs:no_elements(S)) end,</span>
<span class="bold_code">P = sofs:partition(SetFun, Ss),</span>
<span class="bold_code">sofs:to_external(P).</span>
[[[a],[b]],[[c,d],[e,f]]]</pre></div>
</p></div>
<p><a name="partition-3"></a><span class="bold_code">partition(SetFun, Set1, Set2) -> {Set3, Set4}</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">SetFun = <span class="bold_code"><a href="#type-set_fun">set_fun()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code">Set1 = Set2 = Set3 = Set4 = <span class="bold_code"><a href="#type-a_set">a_set()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
</div></p>
<div class="REFBODY"><p>
<p>Returns a pair of sets that, regarded as constituting a
set, forms a <span class="bold_code"><a href="#partition">partition</a></span> 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>
<div class="example"><pre>
1> <span class="bold_code">R1 = sofs:relation([{1,a},{2,b},{3,c}]),</span>
<span class="bold_code">S = sofs:set([2,4,6]),</span>
<span class="bold_code">{R2,R3} = sofs:partition(1, R1, S),</span>
<span class="bold_code">{sofs:to_external(R2),sofs:to_external(R3)}.</span>
{[{2,b}],[{1,a},{3,c}]}</pre></div>
<p><span class="code">partition(F, S1, S2)</span> is equivalent to
<span class="code">{restriction(F, S1, S2),
drestriction(F, S1, S2)}</span>.</p>
</p></div>
<p><a name="partition_family-2"></a><span class="bold_code">partition_family(SetFun, Set) -> Family</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">Family = <span class="bold_code"><a href="#type-family">family()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code">SetFun = <span class="bold_code"><a href="#type-set_fun">set_fun()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code">Set = <span class="bold_code"><a href="#type-a_set">a_set()</a></span></span></div>
</div></p>
<div class="REFBODY"><p>
<p>Returns the <span class="bold_code"><a href="#family">family</a></span>
Family where the indexed set is
a <span class="bold_code"><a href="#partition">partition</a></span>
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 <span class="bold_code"><a href="#equivalence_class">equivalence
class</a></span>.</p>
<div class="example"><pre>
1> <span class="bold_code">S = sofs:relation([{a,a,a,a},{a,a,b,b},{a,b,b,b}]),</span>
<span class="bold_code">SetFun = {external, fun({A,_,C,_}) -> {A,C} end},</span>
<span class="bold_code">F = sofs:partition_family(SetFun, S),</span>
<span class="bold_code">sofs:to_external(F).</span>
[{{a,a},[{a,a,a,a}]},{{a,b},[{a,a,b,b},{a,b,b,b}]}]</pre></div>
</p></div>
<p><a name="product-1"></a><span class="bold_code">product(TupleOfSets) -> Relation</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">Relation = <span class="bold_code"><a href="#type-relation">relation()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code">TupleOfSets = <span class="bold_code"><a href="#type-tuple_of">tuple_of</a></span>(<span class="bold_code"><a href="#type-a_set">a_set()</a></span>)</span></div>
</div></p>
<div class="REFBODY"><p>
<p>Returns the <span class="bold_code"><a href="#Cartesian_product_tuple">Cartesian
product</a></span> of the non-empty tuple of sets
TupleOfSets. If (x[1], ..., x[n]) is
an element of the n-ary relation Relation, then
x[i] is drawn from element i
of TupleOfSets.</p>
<div class="example"><pre>
1> <span class="bold_code">S1 = sofs:set([a,b]),</span>
<span class="bold_code">S2 = sofs:set([1,2]),</span>
<span class="bold_code">S3 = sofs:set([x,y]),</span>
<span class="bold_code">P3 = sofs:product({S1,S2,S3}),</span>
<span class="bold_code">sofs:to_external(P3).</span>
[{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></div>
</p></div>
<p><a name="product-2"></a><span class="bold_code">product(Set1, Set2) -> BinRel</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">BinRel = <span class="bold_code"><a href="#type-binary_relation">binary_relation()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code">Set1 = Set2 = <span class="bold_code"><a href="#type-a_set">a_set()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
</div></p>
<div class="REFBODY"><p>
<p>Returns the <span class="bold_code"><a href="#Cartesian_product">Cartesian
product</a></span> of Set1
and Set2.</p>
<div class="example"><pre>
1> <span class="bold_code">S1 = sofs:set([1,2]),</span>
<span class="bold_code">S2 = sofs:set([a,b]),</span>
<span class="bold_code">R = sofs:product(S1, S2),</span>
<span class="bold_code">sofs:to_external(R).</span>
[{1,a},{1,b},{2,a},{2,b}]</pre></div>
<p><span class="code">product(S1, S2)</span> is equivalent to
<span class="code">product({S1, S2})</span>.</p>
</p></div>
<p><a name="projection-2"></a><span class="bold_code">projection(SetFun, Set1) -> Set2</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">SetFun = <span class="bold_code"><a href="#type-set_fun">set_fun()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code">Set1 = Set2 = <span class="bold_code"><a href="#type-a_set">a_set()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
</div></p>
<div class="REFBODY"><p>
<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 >= 1 and
Set1 is a relation, then the returned set is
the <span class="bold_code"><a href="#projection">projection</a></span> of
Set1 onto coordinate i.</p>
<div class="example"><pre>
1> <span class="bold_code">S1 = sofs:from_term([{1,a},{2,b},{3,a}]),</span>
<span class="bold_code">S2 = sofs:projection(2, S1),</span>
<span class="bold_code">sofs:to_external(S2).</span>
[a,b]</pre></div>
</p></div>
<p><a name="range-1"></a><span class="bold_code">range(BinRel) -> Set</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">BinRel = <span class="bold_code"><a href="#type-binary_relation">binary_relation()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code">Set = <span class="bold_code"><a href="#type-a_set">a_set()</a></span></span></div>
</div></p>
<div class="REFBODY"><p>
<p>Returns the <span class="bold_code"><a href="#range">range</a></span> of the
binary relation BinRel.</p>
<div class="example"><pre>
1> <span class="bold_code">R = sofs:relation([{1,a},{1,b},{2,b},{2,c}]),</span>
<span class="bold_code">S = sofs:range(R),</span>
<span class="bold_code">sofs:to_external(S).</span>
[a,b,c]</pre></div>
</p></div>
<p><a name="relation-1"></a><span class="bold_code">relation(Tuples) -> Relation</span><br><a name="relation-2"></a><span class="bold_code">relation(Tuples, Type) -> Relation</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">N = integer()</span></div>
<div class="REFTYPES"><span class="bold_code">Type = N | <span class="bold_code"><a href="#type-type">type()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code">Relation = <span class="bold_code"><a href="#type-relation">relation()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code">Tuples = [tuple()]</span></div>
</div></p>
<div class="REFBODY"><p>
<p>Creates a <span class="bold_code"><a href="#relation">relation</a></span>.
<span class="code">relation(R, T)</span> is equivalent to
<span class="code">from_term(R, T)</span>, if T is
a <span class="bold_code"><a href="#type">type</a></span> and the result is a
relation. If Type is an integer N, then
<span class="code">[{atom, ..., atom}])</span>, 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. <span class="code">relation([])</span> is
equivalent to <span class="code">relation([], 2)</span>.</p>
</p></div>
<p><a name="relation_to_family-1"></a><span class="bold_code">relation_to_family(BinRel) -> Family</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">Family = <span class="bold_code"><a href="#type-family">family()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code">BinRel = <span class="bold_code"><a href="#type-binary_relation">binary_relation()</a></span></span></div>
</div></p>
<div class="REFBODY"><p>
<p>Returns the <span class="bold_code"><a href="#family">family</a></span>
Family such that the index set is equal to
the <span class="bold_code"><a href="#domain">domain</a></span> of the binary
relation BinRel, and Family[i] is
the <span class="bold_code"><a href="#image">image</a></span> of the set of i
under BinRel.</p>
<div class="example"><pre>
1> <span class="bold_code">R = sofs:relation([{b,1},{c,2},{c,3}]),</span>
<span class="bold_code">F = sofs:relation_to_family(R),</span>
<span class="bold_code">sofs:to_external(F).</span>
[{b,[1]},{c,[2,3]}]</pre></div>
</p></div>
<p><a name="relative_product-1"></a><span class="bold_code">relative_product(ListOfBinRels) -> BinRel2</span><br><a name="relative_product-2"></a><span class="bold_code">relative_product(ListOfBinRels, BinRel1) -> BinRel2</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">ListOfBinRels = [BinRel, ...]</span></div>
<div class="REFTYPES"><span class="bold_code">BinRel = BinRel1 = BinRel2 = <span class="bold_code"><a href="#type-binary_relation">binary_relation()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
</div></p>
<div class="REFBODY"><p>
<p>If ListOfBinRels is a non-empty list
[R[1], ..., R[n]] of binary relations and
BinRel1
is a binary relation, then BinRel2 is the <span class="bold_code"><a href="#tuple_relative_product">relative product</a></span>
of the ordered set (R[i], ..., R[n]) and
BinRel1.</p>
<p>If BinRel1 is omitted, the relation of equality
between the elements of
the <span class="bold_code"><a href="#Cartesian_product_tuple">Cartesian
product</a></span> of the ranges of R[i],
range R[1] × ... × range R[n],
is used instead (intuitively, nothing is "lost").</p>
<div class="example"><pre>
1> <span class="bold_code">TR = sofs:relation([{1,a},{1,aa},{2,b}]),</span>
<span class="bold_code">R1 = sofs:relation([{1,u},{2,v},{3,c}]),</span>
<span class="bold_code">R2 = sofs:relative_product([TR, R1]),</span>
<span class="bold_code">sofs:to_external(R2).</span>
[{1,{a,u}},{1,{aa,u}},{2,{b,v}}]</pre></div>
<p>Note that <span class="code">relative_product([R1], R2)</span> is
different from <span class="code">relative_product(R1, R2)</span>; the
list of one element is not identified with the element
itself.</p>
</p></div>
<p><span class="bold_code">relative_product(BinRel1, BinRel2) -> BinRel3</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">BinRel1 = BinRel2 = BinRel3 = <span class="bold_code"><a href="#type-binary_relation">binary_relation()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
</div></p>
<div class="REFBODY"><p>
<p><a name="relprod_impl"></a>Returns
the <span class="bold_code"><a href="#relative_product">relative
product</a></span> of the binary relations BinRel1
and BinRel2.</p>
</p></div>
<p><a name="relative_product1-2"></a><span class="bold_code">relative_product1(BinRel1, BinRel2) -> BinRel3</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">BinRel1 = BinRel2 = BinRel3 = <span class="bold_code"><a href="#type-binary_relation">binary_relation()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
</div></p>
<div class="REFBODY"><p>
<p>Returns the <span class="bold_code"><a href="#relative_product">relative
product</a></span> of
the <span class="bold_code"><a href="#converse">converse</a></span> of the
binary relation BinRel1 and the binary
relation BinRel2.</p>
<div class="example"><pre>
1> <span class="bold_code">R1 = sofs:relation([{1,a},{1,aa},{2,b}]),</span>
<span class="bold_code">R2 = sofs:relation([{1,u},{2,v},{3,c}]),</span>
<span class="bold_code">R3 = sofs:relative_product1(R1, R2),</span>
<span class="bold_code">sofs:to_external(R3).</span>
[{a,u},{aa,u},{b,v}]</pre></div>
<p><span class="code">relative_product1(R1, R2)</span> is equivalent to
<span class="code">relative_product(converse(R1), R2)</span>.</p>
</p></div>
<p><a name="restriction-2"></a><span class="bold_code">restriction(BinRel1, Set) -> BinRel2</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">BinRel1 = BinRel2 = <span class="bold_code"><a href="#type-binary_relation">binary_relation()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
<div class="REFTYPES"><span class="bold_code">Set = <span class="bold_code"><a href="#type-a_set">a_set()</a></span></span></div>
</div></p>
<div class="REFBODY"><p>
<p>Returns the <span class="bold_code"><a href="#restriction">restriction</a></span> of
the binary relation BinRel1
to Set.</p>
<div class="example"><pre>
1> <span class="bold_code">R1 = sofs:relation([{1,a},{2,b},{3,c}]),</span>
<span class="bold_code">S = sofs:set([1,2,4]),</span>
<span class="bold_code">R2 = sofs:restriction(R1, S),</span>
<span class="bold_code">sofs:to_external(R2).</span>
[{1,a},{2,b}]</pre></div>
</p></div>
<p><a name="restriction-3"></a><span class="bold_code">restriction(SetFun, Set1, Set2) -> Set3</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">SetFun = <span class="bold_code"><a href="#type-set_fun">set_fun()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code">Set1 = Set2 = Set3 = <span class="bold_code"><a href="#type-a_set">a_set()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
</div></p>
<div class="REFBODY"><p>
<p>Returns a subset of Set1 containing those
elements that yield an element in Set2 as the
result of applying SetFun.</p>
<div class="example"><pre>
1> <span class="bold_code">S1 = sofs:relation([{1,a},{2,b},{3,c}]),</span>
<span class="bold_code">S2 = sofs:set([b,c,d]),</span>
<span class="bold_code">S3 = sofs:restriction(2, S1, S2),</span>
<span class="bold_code">sofs:to_external(S3).</span>
[{2,b},{3,c}]</pre></div>
</p></div>
<p><a name="set-1"></a><span class="bold_code">set(Terms) -> Set</span><br><a name="set-2"></a><span class="bold_code">set(Terms, Type) -> Set</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">Set = <span class="bold_code"><a href="#type-a_set">a_set()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code">Terms = [term()]</span></div>
<div class="REFTYPES"><span class="bold_code">Type = <span class="bold_code"><a href="#type-type">type()</a></span></span></div>
</div></p>
<div class="REFBODY"><p>
<p>Creates an <span class="bold_code"><a href="#sets_definition">unordered
set</a></span>. <span class="code">set(L, T)</span> is equivalent to
<span class="code">from_term(L, T)</span>, if the result is an unordered
set. If no <span class="bold_code"><a href="#type">type</a></span> is
explicitly given, <span class="code">[atom]</span> is used as type of the set.</p>
</p></div>
<p><a name="specification-2"></a><span class="bold_code">specification(Fun, Set1) -> Set2</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">Fun = <span class="bold_code"><a href="#type-spec_fun">spec_fun()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code">Set1 = Set2 = <span class="bold_code"><a href="#type-a_set">a_set()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
</div></p>
<div class="REFBODY"><p>
<p>Returns the set containing every element
of Set1 for which Fun
returns <span class="code">true</span>. If Fun is a tuple
<span class="code">{external, Fun2}</span>, Fun2 is applied to the
<span class="bold_code"><a href="#external_set">external set</a></span> of
each element, otherwise Fun is applied to each
element.</p>
<div class="example"><pre>
1> <span class="bold_code">R1 = sofs:relation([{a,1},{b,2}]),</span>
<span class="bold_code">R2 = sofs:relation([{x,1},{x,2},{y,3}]),</span>
<span class="bold_code">S1 = sofs:from_sets([R1,R2]),</span>
<span class="bold_code">S2 = sofs:specification(fun sofs:is_a_function/1, S1),</span>
<span class="bold_code">sofs:to_external(S2).</span>
[[{a,1},{b,2}]]</pre></div>
</p></div>
<p><a name="strict_relation-1"></a><span class="bold_code">strict_relation(BinRel1) -> BinRel2</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">BinRel1 = BinRel2 = <span class="bold_code"><a href="#type-binary_relation">binary_relation()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
</div></p>
<div class="REFBODY"><p>
<p>Returns the <span class="bold_code"><a href="#strict_relation">strict
relation</a></span> corresponding to the binary
relation BinRel1.</p>
<div class="example"><pre>
1> <span class="bold_code">R1 = sofs:relation([{1,1},{1,2},{2,1},{2,2}]),</span>
<span class="bold_code">R2 = sofs:strict_relation(R1),</span>
<span class="bold_code">sofs:to_external(R2).</span>
[{1,2},{2,1}]</pre></div>
</p></div>
<p><a name="substitution-2"></a><span class="bold_code">substitution(SetFun, Set1) -> Set2</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">SetFun = <span class="bold_code"><a href="#type-set_fun">set_fun()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code">Set1 = Set2 = <span class="bold_code"><a href="#type-a_set">a_set()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
</div></p>
<div class="REFBODY"><p>
<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>
<div class="example"><pre>
1> <span class="bold_code">L = [{a,1},{b,2}].</span>
[{a,1},{b,2}]
2> <span class="bold_code">sofs:to_external(sofs:projection(1,sofs:relation(L))).</span>
[a,b]
3> <span class="bold_code">sofs:to_external(sofs:substitution(1,sofs:relation(L))).</span>
[{{a,1},a},{{b,2},b}]
4> <span class="bold_code">SetFun = {external, fun({A,_}=E) -> {E,A} end},</span>
<span class="bold_code">sofs:to_external(sofs:projection(SetFun,sofs:relation(L))).</span>
[{{a,1},a},{{b,2},b}]</pre></div>
<p>The relation of equality between the elements of {a,b,c}:</p>
<div class="example"><pre>
1> <span class="bold_code">I = sofs:substitution(fun(A) -> A end, sofs:set([a,b,c])),</span>
<span class="bold_code">sofs:to_external(I).</span>
[{a,a},{b,b},{c,c}]</pre></div>
<p>Let SetOfSets be a set of sets and BinRel a binary
relation. The function that maps each element Set of
SetOfSets onto the <span class="bold_code"><a href="#image">image</a></span>
of Set under BinRel is returned by this function:</p>
<div class="example"><pre>
images(SetOfSets, BinRel) ->
Fun = fun(Set) -> sofs:image(BinRel, Set) end,
sofs:substitution(Fun, SetOfSets).</pre></div>
<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 <span class="code">images/2</span>, 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>
<div class="example"><pre>
images2(SetOfSets, BinRel) ->
CR = sofs:canonical_relation(SetOfSets),
R = sofs:relative_product1(CR, BinRel),
sofs:relation_to_family(R).</pre></div>
</p></div>
<p><a name="symdiff-2"></a><span class="bold_code">symdiff(Set1, Set2) -> Set3</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">Set1 = Set2 = Set3 = <span class="bold_code"><a href="#type-a_set">a_set()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
</div></p>
<div class="REFBODY"><p>
<p>Returns the <span class="bold_code"><a href="#symmetric_difference">symmetric
difference</a></span> (or the Boolean sum)
of Set1 and Set2.</p>
<div class="example"><pre>
1> <span class="bold_code">S1 = sofs:set([1,2,3]),</span>
<span class="bold_code">S2 = sofs:set([2,3,4]),</span>
<span class="bold_code">P = sofs:symdiff(S1, S2),</span>
<span class="bold_code">sofs:to_external(P).</span>
[1,4]</pre></div>
</p></div>
<p><a name="symmetric_partition-2"></a><span class="bold_code">symmetric_partition(Set1, Set2) -> {Set3, Set4, Set5}</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">Set1 = Set2 = Set3 = Set4 = Set5 = <span class="bold_code"><a href="#type-a_set">a_set()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
</div></p>
<div class="REFBODY"><p>
<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>
</p></div>
<p><a name="to_external-1"></a><span class="bold_code">to_external(AnySet) -> ExternalSet</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">ExternalSet = <span class="bold_code"><a href="#type-external_set">external_set()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code">AnySet = <span class="bold_code"><a href="#type-anyset">anyset()</a></span></span></div>
</div></p>
<div class="REFBODY"><p>
<p>Returns the <span class="bold_code"><a href="#external_set">external
set</a></span> of an atomic, ordered or unordered set.</p>
</p></div>
<p><a name="to_sets-1"></a><span class="bold_code">to_sets(ASet) -> Sets</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">ASet = <span class="bold_code"><a href="#type-a_set">a_set()</a></span> | <span class="bold_code"><a href="#type-ordset">ordset()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code">Sets = <span class="bold_code"><a href="#type-tuple_of">tuple_of</a></span>(AnySet) | [AnySet]</span></div>
<div class="REFTYPES"><span class="bold_code">AnySet = <span class="bold_code"><a href="#type-anyset">anyset()</a></span></span></div>
</div></p>
<div class="REFBODY"><p>
<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>
</p></div>
<p><a name="type-1"></a><span class="bold_code">type(AnySet) -> Type</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">AnySet = <span class="bold_code"><a href="#type-anyset">anyset()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code">Type = <span class="bold_code"><a href="#type-type">type()</a></span></span></div>
</div></p>
<div class="REFBODY"><p>
<p>Returns the <span class="bold_code"><a href="#type">type</a></span> of an
atomic, ordered or unordered set.</p>
</p></div>
<p><a name="union-1"></a><span class="bold_code">union(SetOfSets) -> Set</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">Set = <span class="bold_code"><a href="#type-a_set">a_set()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code">SetOfSets = <span class="bold_code"><a href="#type-set_of_sets">set_of_sets()</a></span></span></div>
</div></p>
<div class="REFBODY"><p>
<p>Returns the <span class="bold_code"><a href="#union_n">union</a></span> of the
set of sets SetOfSets.</p>
</p></div>
<p><a name="union-2"></a><span class="bold_code">union(Set1, Set2) -> Set3</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">Set1 = Set2 = Set3 = <span class="bold_code"><a href="#type-a_set">a_set()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
</div></p>
<div class="REFBODY"><p>
<p>Returns the <span class="bold_code"><a href="#union">union</a></span> of
Set1 and Set2.</p>
</p></div>
<p><a name="union_of_family-1"></a><span class="bold_code">union_of_family(Family) -> Set</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">Family = <span class="bold_code"><a href="#type-family">family()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code">Set = <span class="bold_code"><a href="#type-a_set">a_set()</a></span></span></div>
</div></p>
<div class="REFBODY"><p>
<p>Returns the union of
the <span class="bold_code"><a href="#family">family</a></span> Family.
</p>
<div class="example"><pre>
1> <span class="bold_code">F = sofs:family([{a,[0,2,4]},{b,[0,1,2]},{c,[2,3]}]),</span>
<span class="bold_code">S = sofs:union_of_family(F),</span>
<span class="bold_code">sofs:to_external(S).</span>
[0,1,2,3,4]</pre></div>
</p></div>
<p><a name="weak_relation-1"></a><span class="bold_code">weak_relation(BinRel1) -> BinRel2</span><br><div class="REFBODY">
<p>Types:</p>
<div class="REFTYPES"><span class="bold_code">BinRel1 = BinRel2 = <span class="bold_code"><a href="#type-binary_relation">binary_relation()</a></span></span></div>
<div class="REFTYPES"><span class="bold_code"></span></div>
</div></p>
<div class="REFBODY"><p>
<p>Returns a subset S of the <span class="bold_code"><a href="#weak_relation">weak
relation</a></span> W
corresponding to the binary relation BinRel1.
Let F be the <span class="bold_code"><a href="#field">field</a></span> 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>
<div class="example"><pre>
1> <span class="bold_code">R1 = sofs:relation([{1,1},{1,2},{3,1}]),</span>
<span class="bold_code">R2 = sofs:weak_relation(R1),</span>
<span class="bold_code">sofs:to_external(R2).</span>
[{1,1},{1,2},{2,2},{3,1},{3,3}]</pre></div>
</p></div>
<h3><a name="id221398">See Also</a></h3>
<div class="REFBODY">
<p><span class="bold_code"><a href="dict.html">dict(3)</a></span>,
<span class="bold_code"><a href="digraph.html">digraph(3)</a></span>,
<span class="bold_code"><a href="orddict.html">orddict(3)</a></span>,
<span class="bold_code"><a href="ordsets.html">ordsets(3)</a></span>,
<span class="bold_code"><a href="sets.html">sets(3)</a></span></p>
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