https://algorist.com/algowiki_v2/index.php?action=history&feed=atom&title=TADM2E_2.15
TADM2E 2.15 - Revision history
2026-05-01T01:16:43Z
Revision history for this page on the wiki
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https://algorist.com/algowiki_v2/index.php?title=TADM2E_2.15&diff=464&oldid=prev
Brandon.arnold at 15:36, 5 November 2016
2016-11-05T15:36:27Z
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<td colspan='2' style="background-color: white; color:black; text-align: center;">Revision as of 15:36, 5 November 2016</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; 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>Since <math>a \le b, c \le d \implies ac \le bd</math> it follows that <math>f_1(n) \le {c_3}{g_1(n)}, <del class="diffchange diffchange-inline">f_1</del>(n) \le {c_3}{<del class="diffchange diffchange-inline">g_1</del>(n)} \implies f_1(n) \times f_2(n) \le {c_3}({g_1(n)} \times {g_2(n)})</math>.</div></td><td class='diff-marker'>+</td><td style="color:black; 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>Since <math>a \le b, c \le d \implies ac \le bd</math> it follows that <math>f_1(n) \le {c_3}{g_1(n)}, <ins class="diffchange diffchange-inline">f_2</ins>(n) \le {c_3}{<ins class="diffchange diffchange-inline">g_2</ins>(n)} \implies f_1(n) \times f_2(n) \le {c_3}({g_1(n)} \times {g_2(n)})</math>.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Therefore <math>f_1(n) \times f_2(n) = O(g_1(n) \times g_2(n))</math></div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Therefore <math>f_1(n) \times f_2(n) = O(g_1(n) \times g_2(n))</math></div></td></tr>
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Brandon.arnold
https://algorist.com/algowiki_v2/index.php?title=TADM2E_2.15&diff=462&oldid=prev
Brandon.arnold at 20:36, 4 November 2016
2016-11-04T20:36:55Z
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<td colspan='2' style="background-color: white; color:black; text-align: center;">Revision as of 20:36, 4 November 2016</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l5" >Line 5:</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; 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>Since <math>a \le b, c \le d \implies <del class="diffchange diffchange-inline">ab </del>\le <del class="diffchange diffchange-inline">cd</del></math> it follows that <math>f_1(n) \le {c_3}{g_1(n)}, f_1(n) \le {c_3}{g_1(n)} \implies f_1(n) \times f_2(n) \le {c_3}({g_1(n)} \times {g_2(n)})</math>.</div></td><td class='diff-marker'>+</td><td style="color:black; 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>Since <math>a \le b, c \le d \implies <ins class="diffchange diffchange-inline">ac </ins>\le <ins class="diffchange diffchange-inline">bd</ins></math> it follows that <math>f_1(n) \le {c_3}{g_1(n)}, f_1(n) \le {c_3}{g_1(n)} \implies f_1(n) \times f_2(n) \le {c_3}({g_1(n)} \times {g_2(n)})</math>.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Therefore <math>f_1(n) \times f_2(n) = O(g_1(n) \times g_2(n))</math></div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Therefore <math>f_1(n) \times f_2(n) = O(g_1(n) \times g_2(n))</math></div></td></tr>
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Brandon.arnold
https://algorist.com/algowiki_v2/index.php?title=TADM2E_2.15&diff=459&oldid=prev
Brandon.arnold at 20:31, 4 November 2016
2016-11-04T20:31:12Z
<p></p>
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<td colspan='2' style="background-color: white; color:black; text-align: center;">← Older revision</td>
<td colspan='2' style="background-color: white; color:black; text-align: center;">Revision as of 20:31, 4 November 2016</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l5" >Line 5:</td>
<td colspan="2" class="diff-lineno">Line 5:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; 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>Since <math>a \le b, c \le d \implies <del class="diffchange diffchange-inline">a + b </del>\le <del class="diffchange diffchange-inline">c + d</del></math> it follows that <math>f_1(n) \le {c_3}{g_1(n)}, f_1(n) \le {c_3}{g_1(n)} \implies f_1(n) <del class="diffchange diffchange-inline">+ </del>f_2(n) \le {c_3}({g_1(n)} <del class="diffchange diffchange-inline">+ </del>{g_2(n)})</math>.</div></td><td class='diff-marker'>+</td><td style="color:black; 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>Since <math>a \le b, c \le d \implies <ins class="diffchange diffchange-inline">ab </ins>\le <ins class="diffchange diffchange-inline">cd</ins></math> it follows that <math>f_1(n) \le {c_3}{g_1(n)}, f_1(n) \le {c_3}{g_1(n)} \implies f_1(n) <ins class="diffchange diffchange-inline">\times </ins>f_2(n) \le {c_3}({g_1(n)} <ins class="diffchange diffchange-inline">\times </ins>{g_2(n)})</math>.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; 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>Therefore <math>f_1(n) <del class="diffchange diffchange-inline">+ </del>f_2(n) = O(g_1(n) <del class="diffchange diffchange-inline">+</del>g_2(n))</math></div></td><td class='diff-marker'>+</td><td style="color:black; 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>Therefore <math>f_1(n) <ins class="diffchange diffchange-inline">\times </ins>f_2(n) = O(g_1(n) <ins class="diffchange diffchange-inline">\times </ins>g_2(n))</math></div></td></tr>
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Brandon.arnold
https://algorist.com/algowiki_v2/index.php?title=TADM2E_2.15&diff=458&oldid=prev
Brandon.arnold: Created page with "Choose <math>c_1</math> to satisfy <math>f_1(n) \le {c_1}{g_1(n)}</math> for all <math>n \gt n_{1,0}</math> and <math>c_2</math> to satisfy <math>f_2(n) \le {c_2}{g_2(n)}</mat..."
2016-11-04T20:25:32Z
<p>Created page with "Choose <math>c_1</math> to satisfy <math>f_1(n) \le {c_1}{g_1(n)}</math> for all <math>n \gt n_{1,0}</math> and <math>c_2</math> to satisfy <math>f_2(n) \le {c_2}{g_2(n)}</mat..."</p>
<p><b>New page</b></p><div>Choose <math>c_1</math> to satisfy <math>f_1(n) \le {c_1}{g_1(n)}</math> for all <math>n \gt n_{1,0}</math> and <math>c_2</math> to satisfy <math>f_2(n) \le {c_2}{g_2(n)}</math> for all <math>n \gt n_{2,0}</math>.<br />
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Note that <math>c_1</math> and <math>c_2</math> above can be substituted with <math>c_3</math> such that <math>c_3 = max(c_1, c_2)</math> and the conditions still hold.<br />
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Since <math>a \le b, c \le d \implies a + b \le c + d</math> it follows that <math>f_1(n) \le {c_3}{g_1(n)}, f_1(n) \le {c_3}{g_1(n)} \implies f_1(n) + f_2(n) \le {c_3}({g_1(n)} + {g_2(n)})</math>.<br />
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Therefore <math>f_1(n) + f_2(n) = O(g_1(n) +g_2(n))</math></div>
Brandon.arnold