]>
Commit | Line | Data |
---|---|---|
7c673cae FG |
1 | <html> |
2 | <head> | |
3 | <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> | |
4 | <title>Sets</title> | |
5 | <link rel="stylesheet" href="../../../../../../doc/src/boostbook.css" type="text/css"> | |
6 | <meta name="generator" content="DocBook XSL Stylesheets V1.74.0"> | |
7 | <link rel="home" href="../../index.html" title="Chapter 1. Boost.Icl"> | |
8 | <link rel="up" href="../semantics.html" title="Semantics"> | |
9 | <link rel="prev" href="../semantics.html" title="Semantics"> | |
10 | <link rel="next" href="maps.html" title="Maps"> | |
11 | </head> | |
12 | <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> | |
13 | <table cellpadding="2" width="100%"><tr> | |
14 | <td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td> | |
15 | <td align="center"><a href="../../../../../../index.html">Home</a></td> | |
16 | <td align="center"><a href="../../../../../libraries.htm">Libraries</a></td> | |
17 | <td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> | |
18 | <td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> | |
19 | <td align="center"><a href="../../../../../../more/index.htm">More</a></td> | |
20 | </tr></table> | |
21 | <hr> | |
22 | <div class="spirit-nav"> | |
23 | <a accesskey="p" href="../semantics.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../semantics.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="maps.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> | |
24 | </div> | |
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"><</span><span class="identifier">S</span><span class="special">>::</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"><</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> | |
55 | <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> | |
56 | <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> | |
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">&,</span> | |
68 | <span class="special">&=</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"><</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> | |
74 | <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> | |
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"><</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> | |
91 | <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> | |
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">&</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">&</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">&</span></code> | |
121 | 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> | |
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"><</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> | |
131 | <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> | |
132 | <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> | |
133 | <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> | |
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">+,&</span> <span class="special">&,+</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">&,-</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"><</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> | |
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"><</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> | |
197 | <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> | |
198 | </pre> | |
199 | <p> | |
200 | </p> | |
201 | <p> | |
202 | ||
203 | </p> | |
204 | <pre class="programlisting"> <span class="special">+,&</span> <span class="special">&,+</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"><</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> | |
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 © 2007 -2010 Joachim Faulhaber<br>Copyright © 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> | |
239 | </tr></table> | |
240 | <hr> | |
241 | <div class="spirit-nav"> | |
242 | <a accesskey="p" href="../semantics.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../semantics.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="maps.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> | |
243 | </div> | |
244 | </body> | |
245 | </html> |