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<p>Transciber’s notes:</p>
<ul>
<li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>  </mtext><mo>⟺</mo><mtext>  </mtext></mrow><annotation encoding="application/x-tex">\iff</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.549em;vertical-align:-0.024em;"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">⟺</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span></span></span> is “if and only if”</li>
<li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>  </mtext><mo>⟹</mo><mtext>  </mtext></mrow><annotation encoding="application/x-tex">\implies</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.549em;vertical-align:-0.024em;"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span></span></span> is “implies”</li>
<li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∀</mi></mrow><annotation encoding="application/x-tex">\forall</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord">∀</span></span></span></span> is “for all”</li>
<li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∃</mi></mrow><annotation encoding="application/x-tex">\exists</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord">∃</span></span></span></span> is “exists”</li>
</ul>
<h1 id="big-ohpart-i">Big-Oh–Part I</h1>
<p>CMPT 225</p>
<h2 id="slide-1">Slide 1</h2>
<ul>
<li>Recall: if <code class="language-plaintext highlighter-rouge">f,g</code> are functions <code class="language-plaintext highlighter-rouge">f:N-&gt;N</code>,<code class="language-plaintext highlighter-rouge">g:N-&gt;N</code></li>
<li>f is <code class="language-plaintext highlighter-rouge">O(g)</code> means there are constants <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>n</mi><mn>0</mn></msub><mo separator="true">,</mo><mi>c</mi><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">n_{0},c &gt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335400000000001em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span>
s.t. for every <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>&gt;</mo><msub><mi>n</mi><mn>0</mn></msub><mo separator="true">,</mo><mi>f</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>≤</mo><mtext>c.g</mtext><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">n&gt;n_{0},f(n) \leq \text{c.g}(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">c.g</span></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></li>
<li>that is for all but finitely many “small” values of n: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>≤</mo><mi>c</mi><mi>g</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(n) \leq cg(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></li>
<li>or f grows no faster than g (asymtotically)</li>
<li>we typically write <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mi>O</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(n) = O(g(n))</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">))</span></span></span></span></li>
</ul>
<h2 id="consider-fngn">Consider <code class="language-plaintext highlighter-rouge">f(n)</code>,<code class="language-plaintext highlighter-rouge">g(n)</code>:</h2>
<p>Claim: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mi>O</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(n) = O(g(n))</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">))</span></span></span></span></p>
<p>Why:</p>
<ul>
<li>When <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>&gt;</mo><mi>g</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(n) &gt; g(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> for all n?</li>
<li>But <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>&lt;</mo><mn>2</mn><mi>g</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(n) &lt; 2g(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> for almost all values of <code class="language-plaintext highlighter-rouge">n</code>.</li>
<li>choose <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>n</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">n_{0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> to “exclude” the real values of <code class="language-plaintext highlighter-rouge">n</code>.</li>
<li>Now: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>&lt;</mo><mn>2</mn><mi>g</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(n) &lt; 2g(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> for all <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>&gt;</mo><msub><mi>n</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">n&gt;n_{0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></li>
<li>So: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mi>O</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(n) = O(g(n))</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">))</span></span></span></span></li>
</ul>
<h2 id="also">Also:</h2>
<p>(A graph a cannot transcribe because the notation doesn’t make any sense. All future graphs have complexity on the y axis and the value of n [input to the function] as the x asxis.)</p>
<p>Claim: f(n) is <strong>not</strong> O(g(n))</p>
<p>However large we choose <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi><mo separator="true">,</mo><msub><mi>n</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">c,n_{0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathnormal">c</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, there will be some k (larger than <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>n</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">n_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>) s.d.: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>&gt;</mo><mi>k</mi><mtext>  </mtext><mo>⟹</mo><mtext>  </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>&gt;</mo><mi>c</mi><mi>g</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">n &gt; k \implies f(n) &gt; c g(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.71844em;vertical-align:-0.024em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mat
<h2 id="consider-o1-ie-gn1">Consider <code class="language-plaintext highlighter-rouge">O(1)</code> (i.e. <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">g(n)=1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span>)</h2>
<p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mi>O</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(n) = O(1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord">1</span><span class="mclose">)</span></span></span></span> if <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∃</mi><msub><mi>n</mi><mn>0</mn></msub><mo separator="true">,</mo><mi>c</mi><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\exists n_{0},c &gt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord">∃</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span> s.t. <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∀</mi><mi>n</mi><mo>&gt;</mo><msub><mi>n</mi><mn>0</mn></msub><mtext>  </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>≤</mo><mi>c</mi><mo>×</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\forall n&gt;n_{0} \space \space f(n) \leq c\times 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.73354em;vertical-align:-0.0391em;"></span><span class="mord">∀</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></
<p>(Another graph that makes no Goddamn sense whatsoever.)</p>
<ul>
<li>for every <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>&gt;</mo><msub><mi>n</mi><mn>0</mn></msub><mo separator="true">,</mo><mi>f</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>&lt;</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">n&gt;n_{0}, f(n)&lt;c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">c</span></span></span></span> &lt;– &gt;–&gt;</li>
<li>so, f(n) grwos no faster than a constant</li>
<li>so f(n) is asymptotically bounded by a constant</li>
</ul>
<h2 id="the-constant-does-not-matter">The constant does not matter</h2>
<p>Fact: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mi>O</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mtext>  </mtext><mo>⟺</mo><mtext>  </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mi>O</mi><mo stretchy="false">(</mo><msup><mtext>10</mtext><mn>27</mn></msup><mo stretchy="false">)</mo><mtext>  </mtext><mo>⟺</mo><mtext>  </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mi>O</mi><mo stretchy="false">(</mo><mfrac><mn>1</mn><msup><mtext>10</mtext><mn>27</mn></msup></mfrac><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(n) = O(1) \iff f(n)=O(\text{10}^{27}) \iff f(n) = O(\frac{1}{\text{10}^{27}})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">⟺</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.0984479999999999em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord text"><span class="mord">10</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.848448em;"><span style="top:-3.09734em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">27</span></span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">⟺</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mopen n
<p>Suppose: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mi>O</mi><mo stretchy="false">(</mo><msup><mtext>10</mtext><mn>27</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(n) = O(\text{10}^{27})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.0984479999999999em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord text"><span class="mord">10</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.848448em;"><span style="top:-3.09734em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">27</span></span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> *</p>
<p>Claim: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> is also <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mfrac><mn>1</mn><msup><mtext>10</mtext><mn>27</mn></msup></mfrac><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(\frac{1}{\text{10}^{27}})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">10</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7518971428571428em;"><span style="top:-2.791582857142857em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">27</span></span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">)</span></span></span></span></p>
<p>* means <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∃</mi><msub><mi>n</mi><mn>0</mn></msub><mo separator="true">,</mo><mi>c</mi><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\exists n_{0},c &gt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord">∃</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span> s.t. <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>&gt;</mo><msub><mi>n</mi><mn>0</mn></msub><mtext>  </mtext><mo>⟹</mo><mtext>  </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>≤</mo><mi>c</mi><mo>×</mo><msup><mtext>10</mtext><mn>27</mn></msup></mrow><annotation encoding="application/x-tex">n &gt; n_{0} \implies f(n) \leq c\times\text{10}^{27}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.675em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathnormal">c</span><span class="mspace" style="margin-r
<p>Want to show: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∃</mi><msub><mi>n</mi><mn>0</mn></msub><mo separator="true">,</mo><msup><mi>c</mi><mn>1</mn></msup><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\exists n_{0},c^{1}&gt;0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.008548em;vertical-align:-0.19444em;"></span><span class="mord">∃</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span> s.t. <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>&gt;</mo><msub><mi>n</mi><mn>0</mn></msub><mtext>  </mtext><mo>⟹</mo><mtext>  </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>≤</mo><msup><mi>c</mi><mn>1</mn></msup><mo>×</mo><mfrac><mn>1</mn><msup><mtext>10</mtext><mn>27</mn></msup></mfrac></mrow><annotation encoding="application/x-tex">n&gt;n_{0} \implies f(n) \leq c^{1}\times \frac{1}{\text{10}^{27}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.675em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" s
<p>Choose <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>c</mi><mn>1</mn></msup></mrow><annotation encoding="application/x-tex">c^1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span> big enough that <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>c</mi><mn>1</mn></msup><mfrac><mn>1</mn><msup><mtext>10</mtext><mn>27</mn></msup></mfrac><mo>≤</mo><mi>c</mi><mo>×</mo><msup><mtext>10</mtext><mn>27</mn></msup></mrow><annotation encoding="application/x-tex">c^{1} \frac{1}{\text{10}^{27}} \leq c \times \text{10}^{27}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">10</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7518971428571428em;"><span style="top:-2.791582857142857em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">27</span></span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.848448em;vertical-align:0em;"></span><span class="mord"><span class="mord text"><span class="mord">10</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.848448em;"><span style="top:-3.09734em;margin-right:0.05
<p>e.g: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>c</mi><mn>1</mn></msup><mo>=</mo><mi>c</mi><mo>×</mo><msup><mtext>10</mtext><mn>54</mn></msup></mrow><annotation encoding="application/x-tex">c^{1} = c \times \text{10}^{54}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.848448em;vertical-align:0em;"></span><span class="mord"><span class="mord text"><span class="mord">10</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.848448em;"><span style="top:-3.09734em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">54</span></span></span></span></span></span></span></span></span></span></span></span></p>
<p>Then:</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∀</mi><mi>n</mi><mo>&gt;</mo><msub><mi>n</mi><mn>0</mn></msub><mo separator="true">,</mo><mi>f</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>≤</mo><mi>c</mi><mo>×</mo><msup><mtext>10</mtext><mn>27</mn></msup><mspace linebreak="newline"></mspace><mo>≤</mo><mi>c</mi><mo>×</mo><msup><mtext>10</mtext><mrow><mo>−</mo><mn>27</mn></mrow></msup><mo>−</mo><msup><mtext>10</mtext><mn>54</mn></msup><mspace linebreak="newline"></mspace><mo>≤</mo><msup><mi>c</mi><mn>1</mn></msup><mo>−</mo><msup><mtext>10</mtext><mrow><mo>−</mo><mn>27</mn></mrow></msup></mrow><annotation encoding="application/x-tex">
\forall n&gt;n_{0}, f(n) \leq c \times \text{10}^{27} \\
\leq c \times \text{10}^{-27} - \text{10}^{54}\\
\leq c^1 - \text{10}^{-27}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.73354em;vertical-align:-0.0391em;"></span><span class="mord">∀</span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathdefault">n</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">c</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.8641079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord text"><span class="mord">10</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mtight">7</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">c</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.947438em;vertical-align:-0.08333em;"></span><span class="mord"><span class="mord text"><span class="mord">10</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.864108em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">2</span><span class="mord mtight">7</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.8641079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord text"><span class="mord">10</span></s
<p>(54-27=27)</p>
<p>So: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mi>O</mi><mo stretchy="false">(</mo><msup><mtext>10</mtext><mrow><mo>−</mo><mn>27</mn></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(n)=O(\text{10}^{-27})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.0984479999999999em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord text"><span class="mord">10</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.848448em;"><span style="top:-3.09734em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">27</span></span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></p>
<h2 id="asymtotic-notation-eg-big-oh">Asymtotic Notation (e.g. Big-Oh)</h2>
<ul>
<li>Is <em>not</em> “about” algorithm</li>
<li><em>Is</em> a tool for describing (growth of) functions</li>
<li>It is useful for describing functions related to algorithms + data structures, e.g.:
<ul>
<li>minimum or maximum time taken</li>
<li>minimum or maximum space needed</li>
<li>etc.</li>
</ul>
</li>
<li>We use it so often for worst-case time for an algorithm that we often leave implciit a statement like “let T(n) be the max time taken by algorithm A as an input of size as most n.” This statement is essential.</li>
</ul>
<h2 id="ex-complexity-of-palindrome-checking">Ex. <em>Complexity of Palindrome Checking</em></h2>
<ul>
<li>using a stack &amp; queue</li>
<li>Algorithm:
<ol>
<li>Insert all tokens into a stack &amp; a queue</li>
<li>Repeat pop one token; dequeue one token. if different report ‘no’.</li>
</ol>
</li>
<li>size of input = number of symbols or token</li>
<li>each token is (all together O(1)):
<ul>
<li>pushed on the stack, O(1)</li>
<li>enqueued on the queue, O(1)</li>
<li>popped off the stack, O(1)</li>
<li>dequeued from the queue, O(1)</li>
<li>compared to one other token, O(1)</li>
</ul>
</li>
<li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>n tokens</mtext><mtext>  </mtext><mo>⟹</mo><mtext>  </mtext><mi>n</mi><mo>×</mo><mi>O</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{n tokens} \implies n \times O(1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.71844em;vertical-align:-0.024em;"></span><span class="mord text"><span class="mord">n tokens</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord">1</span><span class="mclose">)</span></span></span></span> time in total</li>
<li>So: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mi>n</mi><mo>×</mo><mi>O</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T(n) = n \times O(1) = O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></li>
</ul>
<h2 id="what-does-ntimes-o1--on-mean">What does <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>×</mo><mi>O</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">n\times O(1) = O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> mean?</h2>
<p>It means: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mi>O</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mtext>  </mtext><mo>⟺</mo><mtext>  </mtext><mi>n</mi><mo>×</mo><mi>f</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(n) = O(1) \iff n\times f(n) = O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">⟺</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p>
<p>To see if it is true:</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mi>O</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mtext>  </mtext><mo>⟺</mo><mtext>  </mtext><mi mathvariant="normal">∃</mi><mi>c</mi><mo>&gt;</mo><mn>0</mn><mtext> s.t. </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>&lt;</mo><mi>c</mi><mo separator="true">,</mo><mtext>for any </mtext><mi>n</mi><mo>∈</mo><mi mathvariant="double-struck">Z</mi><mspace linebreak="newline"></mspace><mtext>  </mtext><mo>⟺</mo><mtext>  </mtext><mi mathvariant="normal">∃</mi><mi>c</mi><mo>&gt;</mo><mn>0</mn><mtext> s.t. </mtext><mi>n</mi><mo>×</mo><mi>f</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>&lt;</mo><mi>c</mi><mo>×</mo><mi>n</mi><mo separator="true">,</mo><mtext>for any </mtext><mi>n</mi><mo>∈</mo><mi mathvariant="double-struck">N</mi><mspace linebreak="newline"></mspace><mtext>  </mtext><mo>⟺</mo><mtext>  </mtext><mi>n</mi><mo>×</mo><mi>f</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">
f(n) = O(1) \iff \exists c&gt;0 \space \text{s.t.} \space f(n) &lt; c, \text{for any} \space n \in \Z\\
\iff \exists c&gt;0 \text{ s.t. } n\times f(n) &lt; c\times n, \text{for any } n \in \N\\
\iff n\times f(n) = O(n)
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathdefault">n</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">⟺</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.73354em;vertical-align:-0.0391em;"></span><span class="mord">∃</span><span class="mord mathdefault">c</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">0</span><span class="mspace"> </span><span class="mord text"><span class="mord">s.t.</span></span><span class="mspace"> </span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathdefault">n</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">c</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord text"><span class="mord">for any</span></span><span class="mspace"> </span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68889em;vertical-align:0em;"></span><span class="mord"><span class="mord mathbb">Z</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.549em;vertical-align:-0.024em;"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">⟺</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.73354em;vertical-align:-0.0391em;"></span><span class="mord">∃</span><span class="mord mathdefault">c</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">0</span><span class="mord text"><span class="mord"> s.t. </span></span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span clas
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