https://artofproblemsolving.com/wiki/index.php?title=Operator_inverse&feed=atom&action=history
Operator inverse - Revision history
2024-03-28T20:14:05Z
Revision history for this page on the wiki
MediaWiki 1.31.1
https://artofproblemsolving.com/wiki/index.php?title=Operator_inverse&diff=19855&oldid=prev
Temperal: categories
2007-11-23T15:40:52Z
<p>categories</p>
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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 15:40, 23 November 2007</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l11" >Line 11:</td>
<td colspan="2" class="diff-lineno">Line 11:</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>If the operation <math>G</math> is [[associative]] and an element has both a right and left inverse, these two inverses are equal.</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>If the operation <math>G</math> is [[associative]] and an element has both a right and left inverse, these two inverses are equal.</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;"><div>===Proof===</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>===Proof===</div></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>Let <math>g</math> be the element with left inverse <math>h</math> and right inverse <math>h'</math>, so <math>G(h, g) = G(g, h') = e</math>.  Then <math>G(G(h, g), h') = G(e, h') = h'</math>, by the properties of <math>e</math>.  But by associativity, <math><del class="diffchange diffchange-inline">\displaystyle </del>G(G(h, g), h') = G(h, G(g, h')) = G(h, e) = h</math>, so we do indeed have <math>h = h'</math>.</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>Let <math>g</math> be the element with left inverse <math>h</math> and right inverse <math>h'</math>, so <math>G(h, g) = G(g, h') = e</math>.  Then <math>G(G(h, g), h') = G(e, h') = h'</math>, by the properties of <math>e</math>.  But by associativity, <math>G(G(h, g), h') = G(h, G(g, h')) = G(h, e) = h</math>, so we do indeed have <math>h = h'</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="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>===Corollary===</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>===Corollary===</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;"><div>If the operation <math>G</math> is associative, inverses are unique.</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>If the operation <math>G</math> is associative, inverses are unique.</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;">[[Category:Abstract algebra]]</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;">[[Category:Definition]]</ins></div></td></tr>
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Temperal
https://artofproblemsolving.com/wiki/index.php?title=Operator_inverse&diff=19852&oldid=prev
Temperal: Operator inversion moved to Operator inverse: actually, that wording was terrible too
2007-11-23T15:39:23Z
<p><a href="/wiki/index.php/Operator_inversion" class="mw-redirect" title="Operator inversion">Operator inversion</a> moved to <a href="/wiki/index.php/Operator_inverse" title="Operator inverse">Operator inverse</a>: actually, that wording was terrible too</p>
<table class="diff diff-contentalign-left" data-mw="interface">
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<td colspan="1" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="1" style="background-color: #fff; color: #222; text-align: center;">Revision as of 15:39, 23 November 2007</td>
</tr><tr><td colspan="2" class="diff-notice" lang="en"><div class="mw-diff-empty">(No difference)</div>
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Temperal
https://artofproblemsolving.com/wiki/index.php?title=Operator_inverse&diff=19849&oldid=prev
Temperal: Inverse with respect to an operation moved to Operator inversion: This wording is terrible
2007-11-23T15:38:09Z
<p><a href="/wiki/index.php/Inverse_with_respect_to_an_operation" class="mw-redirect" title="Inverse with respect to an operation">Inverse with respect to an operation</a> moved to <a href="/wiki/index.php/Operator_inversion" class="mw-redirect" title="Operator inversion">Operator inversion</a>: This wording is terrible</p>
<table class="diff diff-contentalign-left" data-mw="interface">
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<td colspan="1" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="1" style="background-color: #fff; color: #222; text-align: center;">Revision as of 15:38, 23 November 2007</td>
</tr><tr><td colspan="2" class="diff-notice" lang="en"><div class="mw-diff-empty">(No difference)</div>
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Temperal
https://artofproblemsolving.com/wiki/index.php?title=Operator_inverse&diff=9522&oldid=prev
JBL at 18:13, 14 August 2006
2006-08-14T18:13:58Z
<p></p>
<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 18:13, 14 August 2006</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>Suppose we have a [[binary operation]] G on a set S, <math>G:S\times S \to S</math>, and suppose this operation has an [[identity]] e, so that for every <math>g\in S</math> we have <math>G(e, g) = G(g, e) = g</math>.  An '''inverse to ''g<del class="diffchange diffchange-inline">'' ''' </del>under this operation is an element <math>h \in S</math> such that <math>G(h, g) = G(g, h) = e</math>.</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>Suppose we have a [[binary operation]] <ins class="diffchange diffchange-inline"><math></ins>G<ins class="diffchange diffchange-inline"></math> </ins>on a set <ins class="diffchange diffchange-inline"><math></ins>S<ins class="diffchange diffchange-inline"></math></ins>, <math>G:S\times S \to S</math>, and suppose this operation has an [[identity]] <ins class="diffchange diffchange-inline"><math></ins>e<ins class="diffchange diffchange-inline"></math></ins>, so that for every <math>g\in S</math> we have <math>G(e, g) = G(g, e) = g</math>.  An '''inverse to''<ins class="diffchange diffchange-inline">' <math>\mathbf </ins>g<ins class="diffchange diffchange-inline"></math> </ins>under this operation is an element <math>h \in S</math> such that <math>G(h, g) = G(g, h) = e</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="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>Thus, informally, operating by <del class="diffchange diffchange-inline">''</del>g<del class="diffchange diffchange-inline">'' </del>is the "opposite" of operating by <del class="diffchange diffchange-inline">''</del>g<del class="diffchange diffchange-inline">''</del>-inverse.</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>Thus, informally, operating by <ins class="diffchange diffchange-inline"><math></ins>g<ins class="diffchange diffchange-inline"></math> </ins>is the "opposite" of operating by <ins class="diffchange diffchange-inline"><math></ins>g<ins class="diffchange diffchange-inline"></math></ins>-inverse.</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;"></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>If our operation is not [[commutative]], we can talk separately about ''left inverses'' and ''right inverses''.  A left inverse of g would be some h such that <math>G(h, g) = e</math>, while a right inverse would be some h such that <math>G(g, h) = e</math>.</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>If our operation is not [[commutative]], we can talk separately about ''left inverses'' and ''right inverses''.  A left inverse of <ins class="diffchange diffchange-inline"><math></ins>g<ins class="diffchange diffchange-inline"></math> </ins>would be some <ins class="diffchange diffchange-inline"><math></ins>h<ins class="diffchange diffchange-inline"></math> </ins>such that <math>G(h, g) = e</math>, while a right inverse would be some <ins class="diffchange diffchange-inline"><math></ins>h<ins class="diffchange diffchange-inline"></math> </ins>such that <math>G(g, h) = e</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="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>==Uniqueness (under appropriate conditions)==</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>==Uniqueness (under appropriate conditions)==</div></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>If the operation G is [[associative]] and an element has both a right and left inverse, these two inverses are equal.</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>If the operation <ins class="diffchange diffchange-inline"><math></ins>G<ins class="diffchange diffchange-inline"></math> </ins>is [[associative]] and an element has both a right and left inverse, these two inverses are equal.</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;"><div>===Proof===</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>===Proof===</div></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>Let g be the element with left inverse h and right inverse h', so <math>G(h, g) = G(g, h') = e</math>.  Then <math>G(G(h, g), h') = G(e, h') = h'</math>, by the properties of e.  But by associativity, <math>\displaystyle G(G(h, g), h') = G(h, G(g, h')) = G(h, e) = h</math>, so we do indeed have <math>h = h'</math>.</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>Let <ins class="diffchange diffchange-inline"><math></ins>g<ins class="diffchange diffchange-inline"></math> </ins>be the element with left inverse <ins class="diffchange diffchange-inline"><math></ins>h<ins class="diffchange diffchange-inline"></math> </ins>and right inverse <ins class="diffchange diffchange-inline"><math></ins>h'<ins class="diffchange diffchange-inline"></math></ins>, so <math>G(h, g) = G(g, h') = e</math>.  Then <math>G(G(h, g), h') = G(e, h') = h'</math>, by the properties of <ins class="diffchange diffchange-inline"><math></ins>e<ins class="diffchange diffchange-inline"></math></ins>.  But by associativity, <math>\displaystyle G(G(h, g), h') = G(h, G(g, h')) = G(h, e) = h</math>, so we do indeed have <math>h = h'</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="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>===Corollary===</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>===Corollary===</div></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>If the operation G is associative, inverses are unique.</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>If the operation <ins class="diffchange diffchange-inline"><math></ins>G<ins class="diffchange diffchange-inline"></math> </ins>is associative, inverses are unique.</div></td></tr>
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JBL
https://artofproblemsolving.com/wiki/index.php?title=Operator_inverse&diff=7881&oldid=prev
Inscrutableroot: proofreading
2006-07-18T17:09:23Z
<p>proofreading</p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 17:09, 18 July 2006</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;"></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;"></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>If our operation is not [[commutative]], we can talk separately about ''left inverses'' and ''right inverses''.  A left inverse of g would be some h such that <math>G(h, g) = e</math> while a right inverse would be some h such that <math>G(g, h) = e</math>.</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>If our operation is not [[commutative]], we can talk separately about ''left inverses'' and ''right inverses''.  A left inverse of g would be some h such that <math>G(h, g) = e</math><ins class="diffchange diffchange-inline">, </ins>while a right inverse would be some h such that <math>G(g, h) = e</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="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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Inscrutableroot
https://artofproblemsolving.com/wiki/index.php?title=Operator_inverse&diff=7868&oldid=prev
JBL at 13:13, 18 July 2006
2006-07-18T13:13:53Z
<p></p>
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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 13:13, 18 July 2006</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>Suppose we have a [[binary operation]] G on a set S, <math>G:S\times S \to S</math>, and suppose this operation has an [[identity]] e, so that for every <math>g\in S</math> we have <math>G(e, g) = G(g, e) = g</math>.  An '''inverse to g''' under this operation is an element <math>h \in S</math> such that <math>G(h, g) = G(g, h) = e</math>.</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>Suppose we have a [[binary operation]] G on a set S, <math>G:S\times S \to S</math>, and suppose this operation has an [[identity]] e, so that for every <math>g\in S</math> we have <math>G(e, g) = G(g, e) = g</math>.  An '''inverse to <ins class="diffchange diffchange-inline">''</ins>g<ins class="diffchange diffchange-inline">'' </ins>''' under this operation is an element <math>h \in S</math> such that <math>G(h, g) = G(g, h) = e</math>.</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> </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> </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 class="diffchange diffchange-inline">Thus, informally, operating by ''g'' is the "opposite" of operating by ''g''-inverse.</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> </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>If our operation is not [[commutative]], we can talk separately about ''left inverses'' and ''right inverses''.  A left inverse of g would be some h such that <math>G(h, g) = e</math> while a right inverse would be some h such that <math>G(g, h) = e</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>If our operation is not [[commutative]], we can talk separately about ''left inverses'' and ''right inverses''.  A left inverse of g would be some h such that <math>G(h, g) = e</math> while a right inverse would be some h such that <math>G(g, h) = e</math>.</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 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>==Uniqueness (under appropriate conditions)==</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>==Uniqueness (under appropriate conditions)==</div></td></tr>
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JBL
https://artofproblemsolving.com/wiki/index.php?title=Operator_inverse&diff=7867&oldid=prev
JBL: Moved from disambiguation page -- if you have a better name for this article, please move it
2006-07-18T13:11:26Z
<p>Moved from disambiguation page -- if you have a better name for this article, please move it</p>
<p><b>New page</b></p><div>Suppose we have a [[binary operation]] G on a set S, <math>G:S\times S \to S</math>, and suppose this operation has an [[identity]] e, so that for every <math>g\in S</math> we have <math>G(e, g) = G(g, e) = g</math>. An '''inverse to g''' under this operation is an element <math>h \in S</math> such that <math>G(h, g) = G(g, h) = e</math>.<br />
<br />
If our operation is not [[commutative]], we can talk separately about ''left inverses'' and ''right inverses''. A left inverse of g would be some h such that <math>G(h, g) = e</math> while a right inverse would be some h such that <math>G(g, h) = e</math>.<br />
<br />
==Uniqueness (under appropriate conditions)==<br />
If the operation G is [[associative]] and an element has both a right and left inverse, these two inverses are equal.<br />
===Proof===<br />
Let g be the element with left inverse h and right inverse h', so <math>G(h, g) = G(g, h') = e</math>. Then <math>G(G(h, g), h') = G(e, h') = h'</math>, by the properties of e. But by associativity, <math>\displaystyle G(G(h, g), h') = G(h, G(g, h')) = G(h, e) = h</math>, so we do indeed have <math>h = h'</math>.<br />
<br />
===Corollary===<br />
If the operation G is associative, inverses are unique.</div>
JBL