https://artofproblemsolving.com/wiki/index.php?title=Boolean_lattice&feed=atom&action=history
Boolean lattice - Revision history
2024-03-28T22:34:24Z
Revision history for this page on the wiki
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https://artofproblemsolving.com/wiki/index.php?title=Boolean_lattice&diff=25084&oldid=prev
I like pie at 02:46, 21 April 2008
2008-04-21T02:46:57Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 02:46, 21 April 2008</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1" >Line 1:</td>
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<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">{{stub}}</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Given any [[set]] <math>S</math>, the '''boolean lattice''' <math>B(S)</math> is a [[partially ordered set]] whose elements are those of <math>\mathcal{P}(S)</math>, the [[power set]] of <math>S</math>, ordered by inclusion (<math>\subset</math>).</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Given any [[set]] <math>S</math>, the '''boolean lattice''' <math>B(S)</math> is a [[partially ordered set]] whose elements are those of <math>\mathcal{P}(S)</math>, the [[power set]] of <math>S</math>, ordered by inclusion (<math>\subset</math>).</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Every boolean lattice is a [[distributive lattice]], and the poset operations [[meet]] and [[join]] correspond to the set operations [[union]] and [[intersection]].</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">{{stub}}</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">Every boolean lattice is a </del>[[<del class="diffchange diffchange-inline">distributive lattice</del>]]<del class="diffchange diffchange-inline">, and the poset operations [[meet]] and [[join]] correspond to the set operations [[union]] and [[intersection]].</del></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>[[<ins class="diffchange diffchange-inline">Category:Definition</ins>]]</div></td></tr>
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I like pie
https://artofproblemsolving.com/wiki/index.php?title=Boolean_lattice&diff=10416&oldid=prev
JBL at 17:42, 24 October 2006
2006-10-24T17:42:14Z
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<table class="diff diff-contentalign-left" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 17:42, 24 October 2006</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l3" >Line 3:</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Given any [[set]] <math>S</math>, the '''boolean lattice''' <math>B(S)</math> is a [[partially ordered set]] whose elements are those of <math>\mathcal{P}(S)</math>, the [[power set]] of <math>S</math>, ordered by inclusion (<math>\subset</math>).</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Given any [[set]] <math>S</math>, the '''boolean lattice''' <math>B(S)</math> is a [[partially ordered set]] whose elements are those of <math>\mathcal{P}(S)</math>, the [[power set]] of <math>S</math>, ordered by inclusion (<math>\subset</math>).</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>When <math>S</math> has a finite number of elements (say <math>|S| = n</math>), the boolean lattice associated with <math>S</math> is usually denoted <math>B_n</math>.  Thus, the set <math>S = \{1, 2, 3\}</math> is associated with the boolean lattice <math>B_3</math> with elements <math>\emptyset, \{1\}, \{2\}, \{3\}, \{1, 2\}, \{1, 3\}, \{2, 3\}</math> and <math>\{1, 2, 3\}</math>, among which <math>\emptyset</math> is smaller than all others, <math>S = \{1, 2, 3\}</math> is larger than all others, and the other six elements satisfy the relations <math>\{1\}, \{2\} \subset \{1,2\}</math>, <math>\{1\}, \{3\} \subset \{1,3\}</math>, <math>\{2\}, \{3\} \subset \{2,3\}</math> and no others.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>When <math>S</math> has a <ins class="diffchange diffchange-inline">[[</ins>finite<ins class="diffchange diffchange-inline">]] </ins>number of elements (say <math>|S| = n</math>), the boolean lattice associated with <math>S</math> is usually denoted <math>B_n</math>.  Thus, the set <math>S = \{1, 2, 3\}</math> is associated with the boolean lattice <math>B_3</math> with elements <math>\emptyset, \{1\}, \{2\}, \{3\}, \{1, 2\}, \{1, 3\}, \{2, 3\}</math> and <math>\{1, 2, 3\}</math>, among which <math>\emptyset</math> is smaller than all others, <math>S = \{1, 2, 3\}</math> is larger than all others, and the other six elements satisfy the relations <math>\{1\}, \{2\} \subset \{1,2\}</math>, <math>\{1\}, \{3\} \subset \{1,3\}</math>, <math>\{2\}, \{3\} \subset \{2,3\}</math> and no others.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The [[Hasse diagram]] for <math>B_3</math> is given below:</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The [[Hasse diagram]] for <math>B_3</math> is given below:</div></td></tr>
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JBL
https://artofproblemsolving.com/wiki/index.php?title=Boolean_lattice&diff=10402&oldid=prev
JBL at 19:38, 23 October 2006
2006-10-23T19:38:19Z
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<p><b>New page</b></p><div>{{stub}}<br />
<br />
Given any [[set]] <math>S</math>, the '''boolean lattice''' <math>B(S)</math> is a [[partially ordered set]] whose elements are those of <math>\mathcal{P}(S)</math>, the [[power set]] of <math>S</math>, ordered by inclusion (<math>\subset</math>).<br />
<br />
When <math>S</math> has a finite number of elements (say <math>|S| = n</math>), the boolean lattice associated with <math>S</math> is usually denoted <math>B_n</math>. Thus, the set <math>S = \{1, 2, 3\}</math> is associated with the boolean lattice <math>B_3</math> with elements <math>\emptyset, \{1\}, \{2\}, \{3\}, \{1, 2\}, \{1, 3\}, \{2, 3\}</math> and <math>\{1, 2, 3\}</math>, among which <math>\emptyset</math> is smaller than all others, <math>S = \{1, 2, 3\}</math> is larger than all others, and the other six elements satisfy the relations <math>\{1\}, \{2\} \subset \{1,2\}</math>, <math>\{1\}, \{3\} \subset \{1,3\}</math>, <math>\{2\}, \{3\} \subset \{2,3\}</math> and no others.<br />
<br />
The [[Hasse diagram]] for <math>B_3</math> is given below:<br />
<br />
{{image}}<br />
<br />
<br />
Every boolean lattice is a [[distributive lattice]], and the poset operations [[meet]] and [[join]] correspond to the set operations [[union]] and [[intersection]].</div>
JBL