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- <title>Sets</title>
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- </div>
- <div class="section">
- <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>
- <h6>
- <a name="boost_icl.semantics.sets.h0"></a>
- <span class="phrase"><a name="boost_icl.semantics.sets.laws_on_set_union"></a></span><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>
- <h6>
- <a name="boost_icl.semantics.sets.h1"></a>
- <span class="phrase"><a name="boost_icl.semantics.sets.laws_on_set_intersection"></a></span><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>
- <h6>
- <a name="boost_icl.semantics.sets.h2"></a>
- <span class="phrase"><a name="boost_icl.semantics.sets.laws_on_set_difference"></a></span><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>
- <h6>
- <a name="boost_icl.semantics.sets.h3"></a>
- <span class="phrase"><a name="boost_icl.semantics.sets.distributivity_laws"></a></span><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>
- <h6>
- <a name="boost_icl.semantics.sets.h4"></a>
- <span class="phrase"><a name="boost_icl.semantics.sets.demorgan_s_law"></a></span><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>
- <h6>
- <a name="boost_icl.semantics.sets.h5"></a>
- <span class="phrase"><a name="boost_icl.semantics.sets.symmetric_difference"></a></span><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>
- </div>
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- <td align="right"><div class="copyright-footer">Copyright © 2007-2010 Joachim
- Faulhaber<br>Copyright © 1999-2006 Cortex Software
- GmbH<p>
- Distributed under the Boost Software License, Version 1.0. (See accompanying
- file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
- </p>
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