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<html>
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<title>Sets</title>
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<div class="section" lang="en">
<div class="titlepage"><div><div><h3 class="title">
<a name="boost_icl.semantics.sets"></a><a class="link" href="sets.html" title="Sets">Sets</a>
</h3></div></div></div>
<p>
For all set types <code class="computeroutput"><span class="identifier">S</span></code> that
are models concept <code class="computeroutput"><span class="identifier">Set</span></code> (<a href="http://www.cplusplus.com/reference/stl/set/" target="_top"><code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">set</span></code>
</a>, <code class="computeroutput"><a class="link" href="../../boost/icl/interval_set.html" title="Class template interval_set">interval_set</a></code>,
<code class="computeroutput"><a class="link" href="../../boost/icl/separate_interval_set.html" title="Class template separate_interval_set">separate_interval_set</a></code>
and <code class="computeroutput"><a class="link" href="../../boost/icl/split_interval_set.html" title="Class template split_interval_set">split_interval_set</a></code>)
most of the well known mathematical <a href="http://en.wikipedia.org/wiki/Algebra_of_sets" target="_top">laws
on sets</a> were successfully checked via LaBatea. The next tables are
giving an overview over the checked laws ordered by operations. If possible,
the laws are formulated with the stronger lexicographical equality (<code class="computeroutput"><span class="keyword">operator</span> <span class="special">==</span></code>)
which implies the law's validity for the weaker element equality <code class="computeroutput"><span class="identifier">is_element_equal</span></code>. Throughout this chapter
we will denote element equality as <code class="computeroutput"><span class="special">=</span><span class="identifier">e</span><span class="special">=</span></code> instead
of <code class="computeroutput"><span class="identifier">is_element_equal</span></code> where
a short notation is advantageous.
</p>
<a name="boost_icl.semantics.sets.laws_on_set_union"></a><h6>
<a name="id1080192"></a>
<a class="link" href="sets.html#boost_icl.semantics.sets.laws_on_set_union">Laws on set union</a>
</h6>
<p>
For the operation <span class="emphasis"><em><span class="bold"><strong>set union</strong></span></em></span>
available as <code class="computeroutput"><span class="keyword">operator</span> <span class="special">+,</span>
<span class="special">+=,</span> <span class="special">|,</span> <span class="special">|=</span></code> and the neutral element <code class="computeroutput"><span class="identifier">identity_element</span><span class="special"><</span><span class="identifier">S</span><span class="special">>::</span><span class="identifier">value</span><span class="special">()</span></code>
which is the empty set <code class="computeroutput"><span class="identifier">S</span><span class="special">()</span></code> these laws hold:
</p>
<pre class="programlisting"><span class="identifier">Associativity</span><span class="special"><</span><span class="identifier">S</span><span class="special">,+,==</span> <span class="special">>:</span> <span class="identifier">S</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">,</span><span class="identifier">c</span><span class="special">;</span> <span class="identifier">a</span><span class="special">+(</span><span class="identifier">b</span><span class="special">+</span><span class="identifier">c</span><span class="special">)</span> <span class="special">==</span> <span class="special">(</span><span class="identifier">a</span><span class="special">+</span><span class="identifier">b</span><span class="special">)+</span><span class="identifier">c</span>
<span class="identifier">Neutrality</span><span class="special"><</span><span class="identifier">S</span><span class="special">,+,==</span> <span class="special">></span> <span class="special">:</span> <span class="identifier">S</span> <span class="identifier">a</span><span class="special">;</span> <span class="identifier">a</span><span class="special">+</span><span class="identifier">S</span><span class="special">()</span> <span class="special">==</span> <span class="identifier">a</span>
<span class="identifier">Commutativity</span><span class="special"><</span><span class="identifier">S</span><span class="special">,+,==</span> <span class="special">>:</span> <span class="identifier">S</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">;</span> <span class="identifier">a</span><span class="special">+</span><span class="identifier">b</span> <span class="special">==</span> <span class="identifier">b</span><span class="special">+</span><span class="identifier">a</span>
</pre>
<p>
</p>
<a name="boost_icl.semantics.sets.laws_on_set_intersection"></a><h6>
<a name="id1080512"></a>
<a class="link" href="sets.html#boost_icl.semantics.sets.laws_on_set_intersection">Laws on
set intersection</a>
</h6>
<p>
For the operation <span class="emphasis"><em><span class="bold"><strong>set intersection</strong></span></em></span>
available as <code class="computeroutput"><span class="keyword">operator</span> <span class="special">&,</span>
<span class="special">&=</span></code> these laws were validated:
</p>
<p>
</p>
<pre class="programlisting"><span class="identifier">Associativity</span><span class="special"><</span><span class="identifier">S</span><span class="special">,&,==</span> <span class="special">>:</span> <span class="identifier">S</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">,</span><span class="identifier">c</span><span class="special">;</span> <span class="identifier">a</span><span class="special">&(</span><span class="identifier">b</span><span class="special">&</span><span class="identifier">c</span><span class="special">)</span> <span class="special">==</span> <span class="special">(</span><span class="identifier">a</span><span class="special">&</span><span class="identifier">b</span><span class="special">)&</span><span class="identifier">c</span>
<span class="identifier">Commutativity</span><span class="special"><</span><span class="identifier">S</span><span class="special">,&,==</span> <span class="special">>:</span> <span class="identifier">S</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">;</span> <span class="identifier">a</span><span class="special">&</span><span class="identifier">b</span> <span class="special">==</span> <span class="identifier">b</span><span class="special">&</span><span class="identifier">a</span>
</pre>
<p>
</p>
<a name="boost_icl.semantics.sets.laws_on_set_difference"></a><h6>
<a name="id1080727"></a>
<a class="link" href="sets.html#boost_icl.semantics.sets.laws_on_set_difference">Laws on set
difference</a>
</h6>
<p>
For set difference there are only these laws. It is not associative and not
commutative. It's neutrality is non symmetrical.
</p>
<p>
</p>
<pre class="programlisting"><span class="identifier">RightNeutrality</span><span class="special"><</span><span class="identifier">S</span><span class="special">,-,==</span> <span class="special">></span> <span class="special">:</span> <span class="identifier">S</span> <span class="identifier">a</span><span class="special">;</span> <span class="identifier">a</span><span class="special">-</span><span class="identifier">S</span><span class="special">()</span> <span class="special">==</span> <span class="identifier">a</span>
<span class="identifier">Inversion</span><span class="special"><</span><span class="identifier">S</span><span class="special">,-,==</span> <span class="special">>:</span> <span class="identifier">S</span> <span class="identifier">a</span><span class="special">;</span> <span class="identifier">a</span> <span class="special">-</span> <span class="identifier">a</span> <span class="special">==</span> <span class="identifier">S</span><span class="special">()</span>
</pre>
<p>
</p>
<p>
Summarized in the next table are laws that use <code class="computeroutput"><span class="special">+</span></code>,
<code class="computeroutput"><span class="special">&</span></code> and <code class="computeroutput"><span class="special">-</span></code>
as a single operation. For all validated laws, the left and right hand sides
of the equations are lexicographically equal, as denoted by <code class="computeroutput"><span class="special">==</span></code> in the cells of the table.
</p>
<p>
</p>
<pre class="programlisting"> <span class="special">+</span> <span class="special">&</span> <span class="special">-</span>
<span class="identifier">Associativity</span> <span class="special">==</span> <span class="special">==</span>
<span class="identifier">Neutrality</span> <span class="special">==</span> <span class="special">==</span>
<span class="identifier">Commutativity</span> <span class="special">==</span> <span class="special">==</span>
<span class="identifier">Inversion</span> <span class="special">==</span>
</pre>
<p>
</p>
<a name="boost_icl.semantics.sets.distributivity_laws"></a><h6>
<a name="id1080985"></a>
<a class="link" href="sets.html#boost_icl.semantics.sets.distributivity_laws">Distributivity
Laws</a>
</h6>
<p>
Laws, like distributivity, that use more than one operation can sometimes
be instantiated for different sequences of operators as can be seen below.
In the two instantiations of the distributivity laws operators <code class="computeroutput"><span class="special">+</span></code> and <code class="computeroutput"><span class="special">&</span></code>
are swapped. So we can have small operator signatures like <code class="computeroutput"><span class="special">+,&</span></code> and <code class="computeroutput"><span class="special">&,+</span></code>
to describe such instantiations, which will be used below. Not all instances
of distributivity laws hold for lexicographical equality. Therefore they
are denoted using a <span class="emphasis"><em>variable</em></span> equality <code class="computeroutput"><span class="special">=</span><span class="identifier">v</span><span class="special">=</span></code>
below.
</p>
<p>
</p>
<pre class="programlisting"> <span class="identifier">Distributivity</span><span class="special"><</span><span class="identifier">S</span><span class="special">,+,&,=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">></span> <span class="special">:</span> <span class="identifier">S</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">,</span><span class="identifier">c</span><span class="special">;</span> <span class="identifier">a</span> <span class="special">+</span> <span class="special">(</span><span class="identifier">b</span> <span class="special">&</span> <span class="identifier">c</span><span class="special">)</span> <span class="special">=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">+</span> <span class="identifier">b</span><span class="special">)</span> <span class="special">&</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">+</span> <span class="identifier">c</span><span class="special">)</span>
<span class="identifier">Distributivity</span><span class="special"><</span><span class="identifier">S</span><span class="special">,&,+,=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">></span> <span class="special">:</span> <span class="identifier">S</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">,</span><span class="identifier">c</span><span class="special">;</span> <span class="identifier">a</span> <span class="special">&</span> <span class="special">(</span><span class="identifier">b</span> <span class="special">+</span> <span class="identifier">c</span><span class="special">)</span> <span class="special">=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">&</span> <span class="identifier">b</span><span class="special">)</span> <span class="special">+</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">&</span> <span class="identifier">c</span><span class="special">)</span>
<span class="identifier">RightDistributivity</span><span class="special"><</span><span class="identifier">S</span><span class="special">,+,-,=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">></span> <span class="special">:</span> <span class="identifier">S</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">,</span><span class="identifier">c</span><span class="special">;</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">+</span> <span class="identifier">b</span><span class="special">)</span> <span class="special">-</span> <span class="identifier">c</span> <span class="special">=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">-</span> <span class="identifier">c</span><span class="special">)</span> <span class="special">+</span> <span class="special">(</span><span class="identifier">b</span> <span class="special">-</span> <span class="identifier">c</span><span class="special">)</span>
<span class="identifier">RightDistributivity</span><span class="special"><</span><span class="identifier">S</span><span class="special">,&,-,=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">></span> <span class="special">:</span> <span class="identifier">S</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">,</span><span class="identifier">c</span><span class="special">;</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">&</span> <span class="identifier">b</span><span class="special">)</span> <span class="special">-</span> <span class="identifier">c</span> <span class="special">=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">-</span> <span class="identifier">c</span><span class="special">)</span> <span class="special">&</span> <span class="special">(</span><span class="identifier">b</span> <span class="special">-</span> <span class="identifier">c</span><span class="special">)</span>
</pre>
<p>
</p>
<p>
The next table shows the relationship between law instances, <a class="link" href="../../index.html#boost_icl.introduction.interval_combining_styles" title="Interval Combining Styles">interval
combining style</a> and the used equality relation.
</p>
<p>
</p>
<pre class="programlisting"> <span class="special">+,&</span> <span class="special">&,+</span>
<span class="identifier">Distributivity</span> <span class="identifier">joining</span> <span class="special">==</span> <span class="special">==</span>
<span class="identifier">separating</span> <span class="special">==</span> <span class="special">==</span>
<span class="identifier">splitting</span> <span class="special">=</span><span class="identifier">e</span><span class="special">=</span> <span class="special">=</span><span class="identifier">e</span><span class="special">=</span>
<span class="special">+,-</span> <span class="special">&,-</span>
<span class="identifier">RightDistributivity</span> <span class="identifier">joining</span> <span class="special">==</span> <span class="special">==</span>
<span class="identifier">separating</span> <span class="special">==</span> <span class="special">==</span>
<span class="identifier">splitting</span> <span class="special">=</span><span class="identifier">e</span><span class="special">=</span> <span class="special">==</span>
</pre>
<p>
</p>
<p>
The table gives an overview over 12 instantiations of the four distributivity
laws and shows the equalities which the instantiations holds for. For instance
<code class="computeroutput"><span class="identifier">RightDistributivity</span></code> with
operator signature <code class="computeroutput"><span class="special">+,-</span></code> instantiated
for <code class="computeroutput"><a class="link" href="../../boost/icl/split_interval_set.html" title="Class template split_interval_set">split_interval_sets</a></code>
holds only for element equality (denoted as <code class="computeroutput"><span class="special">=</span><span class="identifier">e</span><span class="special">=</span></code>):
</p>
<pre class="programlisting"><span class="identifier">RightDistributivity</span><span class="special"><</span><span class="identifier">S</span><span class="special">,+,-,=</span><span class="identifier">e</span><span class="special">=</span> <span class="special">></span> <span class="special">:</span> <span class="identifier">S</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">,</span><span class="identifier">c</span><span class="special">;</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">+</span> <span class="identifier">b</span><span class="special">)</span> <span class="special">-</span> <span class="identifier">c</span> <span class="special">=</span><span class="identifier">e</span><span class="special">=</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">-</span> <span class="identifier">c</span><span class="special">)</span> <span class="special">+</span> <span class="special">(</span><span class="identifier">b</span> <span class="special">-</span> <span class="identifier">c</span><span class="special">)</span>
</pre>
<p>
The remaining five instantiations of <code class="computeroutput"><span class="identifier">RightDistributivity</span></code>
are valid for lexicographical equality (demoted as <code class="computeroutput"><span class="special">==</span></code>)
as well.
</p>
<p>
<a class="link" href="../../index.html#boost_icl.introduction.interval_combining_styles" title="Interval Combining Styles">Interval
combining styles</a> correspond to containers according to
</p>
<pre class="programlisting"><span class="identifier">style</span> <span class="identifier">set</span>
<span class="identifier">joining</span> <span class="identifier">interval_set</span>
<span class="identifier">separating</span> <span class="identifier">separate_interval_set</span>
<span class="identifier">splitting</span> <span class="identifier">split_interval_set</span>
</pre>
<p>
</p>
<p>
Finally there are two laws that combine all three major set operations: De
Mogans Law and Symmetric Difference.
</p>
<a name="boost_icl.semantics.sets.demorgan_s_law"></a><h6>
<a name="id1082080"></a>
<a class="link" href="sets.html#boost_icl.semantics.sets.demorgan_s_law">DeMorgan's Law</a>
</h6>
<p>
De Morgans Law is better known in an incarnation where the unary complement
operation <code class="computeroutput"><span class="special">~</span></code> is used. <code class="computeroutput"><span class="special">~(</span><span class="identifier">a</span><span class="special">+</span><span class="identifier">b</span><span class="special">)</span> <span class="special">==</span>
<span class="special">~</span><span class="identifier">a</span> <span class="special">*</span> <span class="special">~</span><span class="identifier">b</span></code>.
The version below is an adaption for the binary set difference <code class="computeroutput"><span class="special">-</span></code>, which is also called <span class="emphasis"><em><span class="bold"><strong>relative complement</strong></span></em></span>.
</p>
<pre class="programlisting"><span class="identifier">DeMorgan</span><span class="special"><</span><span class="identifier">S</span><span class="special">,+,&,=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">></span> <span class="special">:</span> <span class="identifier">S</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">,</span><span class="identifier">c</span><span class="special">;</span> <span class="identifier">a</span> <span class="special">-</span> <span class="special">(</span><span class="identifier">b</span> <span class="special">+</span> <span class="identifier">c</span><span class="special">)</span> <span class="special">=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">-</span> <span class="identifier">b</span><span class="special">)</span> <span class="special">&</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">-</span> <span class="identifier">c</span><span class="special">)</span>
<span class="identifier">DeMorgan</span><span class="special"><</span><span class="identifier">S</span><span class="special">,&,+,=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">></span> <span class="special">:</span> <span class="identifier">S</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">,</span><span class="identifier">c</span><span class="special">;</span> <span class="identifier">a</span> <span class="special">-</span> <span class="special">(</span><span class="identifier">b</span> <span class="special">&</span> <span class="identifier">c</span><span class="special">)</span> <span class="special">=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">-</span> <span class="identifier">b</span><span class="special">)</span> <span class="special">+</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">-</span> <span class="identifier">c</span><span class="special">)</span>
</pre>
<p>
</p>
<p>
</p>
<pre class="programlisting"> <span class="special">+,&</span> <span class="special">&,+</span>
<span class="identifier">DeMorgan</span> <span class="identifier">joining</span> <span class="special">==</span> <span class="special">==</span>
<span class="identifier">separating</span> <span class="special">==</span> <span class="special">=</span><span class="identifier">e</span><span class="special">=</span>
<span class="identifier">splitting</span> <span class="special">==</span> <span class="special">=</span><span class="identifier">e</span><span class="special">=</span>
</pre>
<p>
</p>
<p>
Again not all law instances are valid for lexicographical equality. The second
instantiations only holds for element equality, if the interval sets are
non joining.
</p>
<a name="boost_icl.semantics.sets.symmetric_difference"></a><h6>
<a name="id1082555"></a>
<a class="link" href="sets.html#boost_icl.semantics.sets.symmetric_difference">Symmetric Difference</a>
</h6>
<p>
</p>
<pre class="programlisting"><span class="identifier">SymmetricDifference</span><span class="special"><</span><span class="identifier">S</span><span class="special">,==</span> <span class="special">></span> <span class="special">:</span> <span class="identifier">S</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">,</span><span class="identifier">c</span><span class="special">;</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">+</span> <span class="identifier">b</span><span class="special">)</span> <span class="special">-</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">&</span> <span class="identifier">b</span><span class="special">)</span> <span class="special">==</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">-</span> <span class="identifier">b</span><span class="special">)</span> <span class="special">+</span> <span class="special">(</span><span class="identifier">b</span> <span class="special">-</span> <span class="identifier">a</span><span class="special">)</span>
</pre>
<p>
</p>
<p>
Finally Symmetric Difference holds for all of icl set types and lexicographical
equality.
</p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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