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<!DOCTYPE html>
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<title>10. 集合 — Python早見帳</title>
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基礎
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発展
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科学技術計算とデータ分析
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10.1. 基本
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10.2. 集合の内包表記
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10.3. 集合間の演算
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<h1>集合</h1>
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10.1. 基本
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10.2. 集合の内包表記
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10.3. 集合間の演算
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<div class="tex2jax_ignore mathjax_ignore section" id="id1">
<h1><span class="section-number">10. </span>集合<a class="headerlink" href="#id1" title="このヘッドラインへのパーマリンク">#</a></h1>
<div class="section" id="id2">
<h2><span class="section-number">10.1. </span>基本<a class="headerlink" href="#id2" title="このヘッドラインへのパーマリンク">#</a></h2>
<p>集合はユニークな（重複のない）要素をまとめて保持する。「値」のない辞書と考えると分かりやすいかもしれない。</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">S</span> <span class="o">=</span> <span class="p">{</span><span class="s1">'東京'</span><span class="p">,</span> <span class="s1">'神奈川'</span><span class="p">,</span> <span class="s1">'千葉'</span><span class="p">,</span> <span class="s1">'埼玉'</span><span class="p">}</span>
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">S</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>{'千葉', '埼玉', '東京', '神奈川'}
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="nb">type</span><span class="p">(</span><span class="n">S</span><span class="p">)</span>
</pre></div>
</div>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>set
</pre></div>
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</div>
</div>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="nb">len</span><span class="p">(</span><span class="n">S</span><span class="p">)</span>
</pre></div>
</div>
</div>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>4
</pre></div>
</div>
</div>
</div>
<p>集合に要素が含まれているかどうか<code class="docutils literal notranslate"><span class="pre">in</span></code>演算子で調べることができる。</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="s1">'東京'</span> <span class="ow">in</span> <span class="n">S</span>
</pre></div>
</div>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>True
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="s1">'栃木'</span> <span class="ow">in</span> <span class="n">S</span>
</pre></div>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>False
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</div>
<p>集合に要素を追加するには、addメソッドを用いる。</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">S</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="s1">'栃木'</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="s1">'栃木'</span> <span class="ow">in</span> <span class="n">S</span>
</pre></div>
</div>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>True
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<p>集合から要素を削除するには、removeメソッドを用いる。</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">S</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="s1">'栃木'</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="s1">'栃木'</span> <span class="ow">in</span> <span class="n">S</span>
</pre></div>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>False
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<p>集合の要素になっていないものをremoveメソッドで取り除こうとすると、エラーになる。</p>
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<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">S</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="s1">'栃木'</span><span class="p">)</span>
</pre></div>
</div>
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<div class="output traceback highlight-ipythontb notranslate"><div class="highlight"><pre><span></span><span class="gt">---------------------------------------------------------------------------</span>
<span class="ne">KeyError</span><span class="g g-Whitespace"> </span>Traceback (most recent call last)
<span class="o">/</span><span class="n">tmp</span><span class="o">/</span><span class="n">ipykernel_2705</span><span class="o">/</span><span class="mf">730782737.</span><span class="n">py</span> <span class="ow">in</span> <span class="o"><</span><span class="n">module</span><span class="o">></span>
<span class="ne">----> </span><span class="mi">1</span> <span class="n">S</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="s1">'栃木'</span><span class="p">)</span>
<span class="ne">KeyError</span>: '栃木'
</pre></div>
</div>
</div>
</div>
<p>集合の要素になっているときだけ削除するには、discardメソッドを用いる。</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">S</span><span class="o">.</span><span class="n">discard</span><span class="p">(</span><span class="s1">'栃木'</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<p>要素が空の集合（空集合）はset関数で作成する（<code class="docutils literal notranslate"><span class="pre">{}</span></code>は辞書となってしまうため、集合とは異なるオブジェクトになってしまう）。</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">S</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
<span class="n">S</span>
</pre></div>
</div>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>set()
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="id3">
<h2><span class="section-number">10.2. </span>集合の内包表記<a class="headerlink" href="#id3" title="このヘッドラインへのパーマリンク">#</a></h2>
<p>九九の３の段を表現する集合オブジェクトを作成してみる。</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">m3</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">10</span><span class="p">):</span>
<span class="n">m3</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">i</span> <span class="o">*</span> <span class="mi">3</span><span class="p">)</span>
<span class="n">m3</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>{0, 3, 6, 9, 12, 15, 18, 21, 24, 27}
</pre></div>
</div>
</div>
</div>
<p>このオブジェクトは、次のように集合の内包表記で構築することもできる。「i times 3 for all i in the range of ten」のように英語読みすると分かりやすいかもしれない。</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">m3n</span> <span class="o">=</span> <span class="p">{</span><span class="n">i</span><span class="o">*</span><span class="mi">3</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">10</span><span class="p">)}</span>
<span class="n">m3n</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>{0, 3, 6, 9, 12, 15, 18, 21, 24, 27}
</pre></div>
</div>
</div>
</div>
<p>また、「0から29の範囲の中で3で割ったあまりが0になる数字の集合」というアイディアに基づいた実装をすると、以下のようになる。</p>
<div class="cell docutils container">
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">m3nn</span> <span class="o">=</span> <span class="p">{</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">30</span><span class="p">)</span> <span class="k">if</span> <span class="n">i</span> <span class="o">%</span> <span class="mi">3</span> <span class="o">==</span> <span class="mi">0</span><span class="p">}</span>
<span class="n">m3nn</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>{0, 3, 6, 9, 12, 15, 18, 21, 24, 27}
</pre></div>
</div>
</div>
</div>
<p>集合の内包表記をネストさせることで、九九で出てくる数字を表す集合を作成できる。</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">M</span> <span class="o">=</span> <span class="p">{</span><span class="n">i</span><span class="o">*</span><span class="n">j</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">10</span><span class="p">)}</span>
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">M</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>{0,
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
12,
14,
15,
16,
18,
20,
21,
24,
25,
27,
28,
30,
32,
35,
36,
40,
42,
45,
48,
49,
54,
56,
63,
64,
72,
81}
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="id4">
<h2><span class="section-number">10.3. </span>集合間の演算<a class="headerlink" href="#id4" title="このヘッドラインへのパーマリンク">#</a></h2>
<p><span class="math notranslate nohighlight">\(1\)</span>以上<span class="math notranslate nohighlight">\(n\)</span>以下の自然数を<span class="math notranslate nohighlight">\(\mathbb{N}_n\)</span>と書くことにする。<span class="math notranslate nohighlight">\(1\)</span>以上<span class="math notranslate nohighlight">\(30\)</span>以下の<span class="math notranslate nohighlight">\(2\)</span>の倍数と<span class="math notranslate nohighlight">\(3\)</span>の倍数を表す集合はそれぞれ、</p>
<div class="math notranslate nohighlight">
\[\begin{split}
M_2 = \{i \in \mathbb{N}_{30} \mid i \equiv 0 \pmod{2} \} \\
M_3 = \{i \in \mathbb{N}_{30} \mid i \equiv 0 \pmod{3} \}
\end{split}\]</div>
<p>と書ける。<span class="math notranslate nohighlight">\(\mathbb{N}_{30}\)</span>と<span class="math notranslate nohighlight">\(M_2\)</span>、<span class="math notranslate nohighlight">\(M_3\)</span>をそれぞれ、集合型のオブジェクト<code class="docutils literal notranslate"><span class="pre">N</span></code>と<code class="docutils literal notranslate"><span class="pre">M2</span></code>、<code class="docutils literal notranslate"><span class="pre">M3</span></code>で表す。</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">N</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">31</span><span class="p">))</span>
<span class="n">M2</span> <span class="o">=</span> <span class="p">{</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">N</span> <span class="k">if</span> <span class="n">i</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">0</span><span class="p">}</span>
<span class="n">M3</span> <span class="o">=</span> <span class="p">{</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">N</span> <span class="k">if</span> <span class="n">i</span> <span class="o">%</span> <span class="mi">3</span> <span class="o">==</span> <span class="mi">0</span><span class="p">}</span>
</pre></div>
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</div>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">M2</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>{2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30}
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">M3</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>{3, 6, 9, 12, 15, 18, 21, 24, 27, 30}
</pre></div>
</div>
</div>
</div>
<p><span class="math notranslate nohighlight">\(M_2\)</span>と<span class="math notranslate nohighlight">\(M_3\)</span>の積集合<span class="math notranslate nohighlight">\(M_2 \cap M_3\)</span>は<code class="docutils literal notranslate"><span class="pre">&</span></code>演算子で求められる。</p>
<div class="cell docutils container">
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">M2</span> <span class="o">&</span> <span class="n">M3</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>{6, 12, 18, 24, 30}
</pre></div>
</div>
</div>
</div>
<p><span class="math notranslate nohighlight">\(M_2\)</span>と<span class="math notranslate nohighlight">\(M_3\)</span>の和集合<span class="math notranslate nohighlight">\(M_2 \cup M_3\)</span>は<code class="docutils literal notranslate"><span class="pre">|</span></code>演算子で求められる。</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">M2</span> <span class="o">|</span> <span class="n">M3</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>{2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30}
</pre></div>
</div>
</div>
</div>
<p><span class="math notranslate nohighlight">\(\mathbb{N}_{30}\)</span>と<span class="math notranslate nohighlight">\(M_2\)</span>の差集合<span class="math notranslate nohighlight">\(\mathbb{N}_{30} \setminus M_2\)</span>は<code class="docutils literal notranslate"><span class="pre">-</span></code>演算子で求められる。</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">N</span> <span class="o">-</span> <span class="n">M2</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>{1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29}
</pre></div>
</div>
</div>
</div>
<p><span class="math notranslate nohighlight">\(M_2\)</span>か<span class="math notranslate nohighlight">\(M_3\)</span>のいずれか一方に含まれる要素を集めた集合は<code class="docutils literal notranslate"><span class="pre">^</span></code>演算子で求められる。</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">M2</span> <span class="o">^</span> <span class="n">M3</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>{2, 3, 4, 8, 9, 10, 14, 15, 16, 20, 21, 22, 26, 27, 28}
</pre></div>
</div>
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<p>2つの集合が部分集合の関係にあるかどうかは、<code class="docutils literal notranslate"><span class="pre"><</span></code>演算子で調べられる。</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">M2</span> <span class="o"><</span> <span class="n">N</span>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>True
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">M2</span> <span class="o"><</span> <span class="n">M3</span>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>False
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<p>2つの集合の要素が全て等しい／等しくないかは、<code class="docutils literal notranslate"><span class="pre">==</span></code>演算子および<code class="docutils literal notranslate"><span class="pre">!=</span></code>演算子で調べられる。</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">X</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">31</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>{2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30}
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">X</span> <span class="o">==</span> <span class="n">M2</span>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>True
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">X</span> <span class="o">!=</span> <span class="n">M2</span>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>False
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">X</span> <span class="o">==</span> <span class="n">M3</span>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>False
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