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| <div class="section boost_icl_semantics_sets" 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="id1085081"></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="id1085402"></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="id1085616"></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="id1085874"></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="id1088062"></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="id1088537"></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> |
| </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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