ceph/src/boost/libs/icl/doc/html/boost_icl/semantics/sets.html

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  25 <div class="section boost_icl_semantics_sets" lang="en">
  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="boost_icl.semantics.sets"></a><a class="link" href="sets.html" title="Sets">Sets</a>
  28 </h3></div></div></div>
  29 <p>
  30         For all set types <code class="computeroutput"><span class="identifier">S</span></code> that
  31         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>
  32         </a>, <code class="computeroutput"><a class="link" href="../../boost/icl/interval_set.html" title="Class template interval_set">interval_set</a></code>,
  33         <code class="computeroutput"><a class="link" href="../../boost/icl/separate_interval_set.html" title="Class template separate_interval_set">separate_interval_set</a></code>
  34         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>)
  35         most of the well known mathematical <a href="http://en.wikipedia.org/wiki/Algebra_of_sets" target="_top">laws
  36         on sets</a> were successfully checked via LaBatea. The next tables are
  37         giving an overview over the checked laws ordered by operations. If possible,
  38         the laws are formulated with the stronger lexicographical equality (<code class="computeroutput"><span class="keyword">operator</span> <span class="special">==</span></code>)
  39         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
  40         we will denote element equality as <code class="computeroutput"><span class="special">=</span><span class="identifier">e</span><span class="special">=</span></code> instead
  41         of <code class="computeroutput"><span class="identifier">is_element_equal</span></code> where
  42         a short notation is advantageous.
  43       </p>
  44 <a name="boost_icl.semantics.sets.laws_on_set_union"></a><h6>
  45 <a name="id1085081"></a>
  46         <a class="link" href="sets.html#boost_icl.semantics.sets.laws_on_set_union">Laws on set union</a>
  47       </h6>
  48 <p>
  49         For the operation <span class="emphasis"><em><span class="bold"><strong>set union</strong></span></em></span>
  50         available as <code class="computeroutput"><span class="keyword">operator</span> <span class="special">+,</span>
  51         <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">&lt;</span><span class="identifier">S</span><span class="special">&gt;::</span><span class="identifier">value</span><span class="special">()</span></code>
  52         which is the empty set <code class="computeroutput"><span class="identifier">S</span><span class="special">()</span></code> these laws hold:
  53 </p>
  54 <pre class="programlisting"><span class="identifier">Associativity</span><span class="special">&lt;</span><span class="identifier">S</span><span class="special">,+,==</span> <span class="special">&gt;:</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>
  55 <span class="identifier">Neutrality</span><span class="special">&lt;</span><span class="identifier">S</span><span class="special">,+,==</span> <span class="special">&gt;</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>
  56 <span class="identifier">Commutativity</span><span class="special">&lt;</span><span class="identifier">S</span><span class="special">,+,==</span> <span class="special">&gt;:</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>
  57 </pre>
  58 <p>
  59       </p>
  60 <a name="boost_icl.semantics.sets.laws_on_set_intersection"></a><h6>
  61 <a name="id1085402"></a>
  62         <a class="link" href="sets.html#boost_icl.semantics.sets.laws_on_set_intersection">Laws on
  63         set intersection</a>
  64       </h6>
  65 <p>
  66         For the operation <span class="emphasis"><em><span class="bold"><strong>set intersection</strong></span></em></span>
  67         available as <code class="computeroutput"><span class="keyword">operator</span> <span class="special">&amp;,</span>
  68         <span class="special">&amp;=</span></code> these laws were validated:
  69       </p>
  70 <p>
  71
  72 </p>
  73 <pre class="programlisting"><span class="identifier">Associativity</span><span class="special">&lt;</span><span class="identifier">S</span><span class="special">,&amp;,==</span> <span class="special">&gt;:</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">&amp;(</span><span class="identifier">b</span><span class="special">&amp;</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">&amp;</span><span class="identifier">b</span><span class="special">)&amp;</span><span class="identifier">c</span>
  74 <span class="identifier">Commutativity</span><span class="special">&lt;</span><span class="identifier">S</span><span class="special">,&amp;,==</span> <span class="special">&gt;:</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">&amp;</span><span class="identifier">b</span> <span class="special">==</span> <span class="identifier">b</span><span class="special">&amp;</span><span class="identifier">a</span>
  75 </pre>
  76 <p>
  77       </p>
  78 <a name="boost_icl.semantics.sets.laws_on_set_difference"></a><h6>
  79 <a name="id1085616"></a>
  80         <a class="link" href="sets.html#boost_icl.semantics.sets.laws_on_set_difference">Laws on set
  81         difference</a>
  82       </h6>
  83 <p>
  84         For set difference there are only these laws. It is not associative and not
  85         commutative. It's neutrality is non symmetrical.
  86       </p>
  87 <p>
  88
  89 </p>
  90 <pre class="programlisting"><span class="identifier">RightNeutrality</span><span class="special">&lt;</span><span class="identifier">S</span><span class="special">,-,==</span> <span class="special">&gt;</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>
  91 <span class="identifier">Inversion</span><span class="special">&lt;</span><span class="identifier">S</span><span class="special">,-,==</span> <span class="special">&gt;:</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>
  92 </pre>
  93 <p>
  94       </p>
  95 <p>
  96         Summarized in the next table are laws that use <code class="computeroutput"><span class="special">+</span></code>,
  97         <code class="computeroutput"><span class="special">&amp;</span></code> and <code class="computeroutput"><span class="special">-</span></code>
  98         as a single operation. For all validated laws, the left and right hand sides
  99         of the equations are lexicographically equal, as denoted by <code class="computeroutput"><span class="special">==</span></code> in the cells of the table.
 100       </p>
 101 <p>
 102
 103 </p>
 104 <pre class="programlisting">                 <span class="special">+</span>    <span class="special">&amp;</span>   <span class="special">-</span>
 105 <span class="identifier">Associativity</span>    <span class="special">==</span>   <span class="special">==</span>
 106 <span class="identifier">Neutrality</span>       <span class="special">==</span>       <span class="special">==</span>
 107 <span class="identifier">Commutativity</span>    <span class="special">==</span>   <span class="special">==</span>
 108 <span class="identifier">Inversion</span>                 <span class="special">==</span>
 109 </pre>
 110 <p>
 111       </p>
 112 <a name="boost_icl.semantics.sets.distributivity_laws"></a><h6>
 113 <a name="id1085874"></a>
 114         <a class="link" href="sets.html#boost_icl.semantics.sets.distributivity_laws">Distributivity
 115         Laws</a>
 116       </h6>
 117 <p>
 118         Laws, like distributivity, that use more than one operation can sometimes
 119         be instantiated for different sequences of operators as can be seen below.
 120         In the two instantiations of the distributivity laws operators <code class="computeroutput"><span class="special">+</span></code> and <code class="computeroutput"><span class="special">&amp;</span></code>
 121         are swapped. So we can have small operator signatures like <code class="computeroutput"><span class="special">+,&amp;</span></code> and <code class="computeroutput"><span class="special">&amp;,+</span></code>
 122         to describe such instantiations, which will be used below. Not all instances
 123         of distributivity laws hold for lexicographical equality. Therefore they
 124         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>
 125         below.
 126       </p>
 127 <p>
 128
 129 </p>
 130 <pre class="programlisting">     <span class="identifier">Distributivity</span><span class="special">&lt;</span><span class="identifier">S</span><span class="special">,+,&amp;,=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">&gt;</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">&amp;</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">&amp;</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">+</span> <span class="identifier">c</span><span class="special">)</span>
 131      <span class="identifier">Distributivity</span><span class="special">&lt;</span><span class="identifier">S</span><span class="special">,&amp;,+,=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">&gt;</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">&amp;</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">&amp;</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">&amp;</span> <span class="identifier">c</span><span class="special">)</span>
 132 <span class="identifier">RightDistributivity</span><span class="special">&lt;</span><span class="identifier">S</span><span class="special">,+,-,=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">&gt;</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>
 133 <span class="identifier">RightDistributivity</span><span class="special">&lt;</span><span class="identifier">S</span><span class="special">,&amp;,-,=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">&gt;</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">&amp;</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">&amp;</span> <span class="special">(</span><span class="identifier">b</span> <span class="special">-</span> <span class="identifier">c</span><span class="special">)</span>
 134 </pre>
 135 <p>
 136       </p>
 137 <p>
 138         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
 139         combining style</a> and the used equality relation.
 140       </p>
 141 <p>
 142
 143 </p>
 144 <pre class="programlisting">                                  <span class="special">+,&amp;</span>    <span class="special">&amp;,+</span>
 145      <span class="identifier">Distributivity</span>  <span class="identifier">joining</span>      <span class="special">==</span>     <span class="special">==</span>
 146                      <span class="identifier">separating</span>   <span class="special">==</span>     <span class="special">==</span>
 147                      <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>
 148
 149                                   <span class="special">+,-</span>    <span class="special">&amp;,-</span>
 150 <span class="identifier">RightDistributivity</span>  <span class="identifier">joining</span>      <span class="special">==</span>     <span class="special">==</span>
 151                      <span class="identifier">separating</span>   <span class="special">==</span>     <span class="special">==</span>
 152                      <span class="identifier">splitting</span>    <span class="special">=</span><span class="identifier">e</span><span class="special">=</span>    <span class="special">==</span>
 153 </pre>
 154 <p>
 155       </p>
 156 <p>
 157         The table gives an overview over 12 instantiations of the four distributivity
 158         laws and shows the equalities which the instantiations holds for. For instance
 159         <code class="computeroutput"><span class="identifier">RightDistributivity</span></code> with
 160         operator signature <code class="computeroutput"><span class="special">+,-</span></code> instantiated
 161         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>
 162         holds only for element equality (denoted as <code class="computeroutput"><span class="special">=</span><span class="identifier">e</span><span class="special">=</span></code>):
 163 </p>
 164 <pre class="programlisting"><span class="identifier">RightDistributivity</span><span class="special">&lt;</span><span class="identifier">S</span><span class="special">,+,-,=</span><span class="identifier">e</span><span class="special">=</span> <span class="special">&gt;</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>
 165 </pre>
 166 <p>
 167         The remaining five instantiations of <code class="computeroutput"><span class="identifier">RightDistributivity</span></code>
 168         are valid for lexicographical equality (demoted as <code class="computeroutput"><span class="special">==</span></code>)
 169         as well.
 170       </p>
 171 <p>
 172         <a class="link" href="../../index.html#boost_icl.introduction.interval_combining_styles" title="Interval Combining Styles">Interval
 173         combining styles</a> correspond to containers according to
 174 </p>
 175 <pre class="programlisting"><span class="identifier">style</span>       <span class="identifier">set</span>
 176 <span class="identifier">joining</span>     <span class="identifier">interval_set</span>
 177 <span class="identifier">separating</span>  <span class="identifier">separate_interval_set</span>
 178 <span class="identifier">splitting</span>   <span class="identifier">split_interval_set</span>
 179 </pre>
 180 <p>
 181       </p>
 182 <p>
 183         Finally there are two laws that combine all three major set operations: De
 184         Mogans Law and Symmetric Difference.
 185       </p>
 186 <a name="boost_icl.semantics.sets.demorgan_s_law"></a><h6>
 187 <a name="id1088062"></a>
 188         <a class="link" href="sets.html#boost_icl.semantics.sets.demorgan_s_law">DeMorgan's Law</a>
 189       </h6>
 190 <p>
 191         De Morgans Law is better known in an incarnation where the unary complement
 192         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>
 193         <span class="special">~</span><span class="identifier">a</span> <span class="special">*</span> <span class="special">~</span><span class="identifier">b</span></code>.
 194         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>.
 195 </p>
 196 <pre class="programlisting"><span class="identifier">DeMorgan</span><span class="special">&lt;</span><span class="identifier">S</span><span class="special">,+,&amp;,=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">&gt;</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">&amp;</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">-</span> <span class="identifier">c</span><span class="special">)</span>
 197 <span class="identifier">DeMorgan</span><span class="special">&lt;</span><span class="identifier">S</span><span class="special">,&amp;,+,=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">&gt;</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">&amp;</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>
 198 </pre>
 199 <p>
 200       </p>
 201 <p>
 202
 203 </p>
 204 <pre class="programlisting">                         <span class="special">+,&amp;</span>     <span class="special">&amp;,+</span>
 205 <span class="identifier">DeMorgan</span>  <span class="identifier">joining</span>        <span class="special">==</span>      <span class="special">==</span>
 206           <span class="identifier">separating</span>     <span class="special">==</span>      <span class="special">=</span><span class="identifier">e</span><span class="special">=</span>
 207           <span class="identifier">splitting</span>      <span class="special">==</span>      <span class="special">=</span><span class="identifier">e</span><span class="special">=</span>
 208 </pre>
 209 <p>
 210       </p>
 211 <p>
 212         Again not all law instances are valid for lexicographical equality. The second
 213         instantiations only holds for element equality, if the interval sets are
 214         non joining.
 215       </p>
 216 <a name="boost_icl.semantics.sets.symmetric_difference"></a><h6>
 217 <a name="id1088537"></a>
 218         <a class="link" href="sets.html#boost_icl.semantics.sets.symmetric_difference">Symmetric Difference</a>
 219       </h6>
 220 <p>
 221
 222 </p>
 223 <pre class="programlisting"><span class="identifier">SymmetricDifference</span><span class="special">&lt;</span><span class="identifier">S</span><span class="special">,==</span> <span class="special">&gt;</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">&amp;</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>
 224 </pre>
 225 <p>
 226       </p>
 227 <p>
 228         Finally Symmetric Difference holds for all of icl set types and lexicographical
 229         equality.
 230       </p>
 231 </div>
 232 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 233 <td align="left"></td>
 234 <td align="right"><div class="copyright-footer">Copyright &#169; 2007 -2010 Joachim Faulhaber<br>Copyright &#169; 1999 -2006 Cortex Software GmbH<p>
 235         Distributed under the Boost Software License, Version 1.0. (See accompanying
 236         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>)
 237       </p>
 238 </div></td>
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