数学 - Python - 集合と位相 - 集合と写像 - 集合間の演算(差、補集合、和集合、共通部分、分配律)

学習環境

• Surface 3 (4G LTE)、Surface 3 タイプ カバー、Surface ペン(端末)
• Windows 10 Pro (OS)
• 数式入力ソフト(TeX, MathML): MathType
• MathML対応ブラウザ: Firefox、Safari
• MathML非対応ブラウザ(Internet Explorer, Microsoft Edge, Google Chrome...)用JavaScript Library: MathJax
• 参考書籍
  • Pythonからはじめる数学入門
  • JavaScriptリファレンス 第6版
  • インタラクティブ・データビジュアライゼーション

集合・位相入門 (松坂 和夫(著)、岩波書店)の第1章(集合と写像)、2(集合間の演算)、練習問題4を取り組んでみる。

4. 1. 
      $\begin{array}{l}A-\left(B\cup C\right)\\ =A\cap {\left(B\cup C\right)}^c\\ =A\cap \left(B^c\cap C^c\right)\\ =A\cap B^c\cap C^c\\ \\ \left(A-B\right)\cap \left(A-C\right)\\ =\left(A\cap B^c\right)\cap \left(A\cap C^c\right)\\ =A\cap B^c\cap A\cap C^c\\ =A\cap B^c\cap C^c\end{array}$
   2. 
      $\begin{array}{l}A-\left(B\cap C\right)\\ =A\cap {\left(B\cap C\right)}^c\\ =A\cap \left(B^c\cup C^c\right)\\ =\left(A\cap B^c\right)\cup \left(A\cup C^c\right)\\ \\ \left(A-B\right)\cup \left(A-C\right)\\ =\left(A\cap B^c\right)\cup \left(A\cap C^c\right)\end{array}$
   3. 
      $\begin{array}{l}\left(A\cup B\right)-C\\ =\left(A\cup B\right)\cap C^c\\ =\left(A\cap C^c\right)\cup \left(B\cap C^c\right)\\ =\left(A-C\right)\cup \left(B-C\right)\end{array}$
   4. 
      $\begin{array}{l}\left(A\cap B\right)-C\\ =\left(A\cap B\right)\cap C^c\\ =A\cap B\cap C^c\\ \\ \left(A-C\right)\cap \left(B-C\right)\\ =\left(A\cap C^c\right)\cap \left(B\cap C^c\right)\\ =A\cap C^c\cap B\cap C^c\\ =A\cap B\cap C^c\end{array}$
   5. 
      $\begin{array}{l}A\cap \left(B-C\right)\\ =A\cap \left(B\cap C^c\right)\\ =A\cap B\cap C^c\\ \\ \left(A\cap B\right)-\left(A\cap C\right)\\ =\left(A\cap B\right)\cap {\left(A\cap C\right)}^c\\ =\left(A\cap B\right)\cap \left(A^c\cup C^c\right)\\ =\left(\left(A\cap B\right)\cap A^c\right)\cup \left(\left(A\cap B\right)\cap C^c\right)\\ =\left(A\cap B\cap A^c\right)\cup \left(A\cap B\cap C^c\right)\\ =\varphi \cup \left(A\cap B\cap C^c\right)\\ =A\cap B\cap C^c\end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
# -*- coding: utf-8 -*-

from matplotlib_venn import venn3_unweighted
import matplotlib.pyplot as plt

from sympy import pprint, FiniteSet, Interval

print('4.')

X = FiniteSet(*range(7))
A = FiniteSet(*range(5))
B = FiniteSet(*range(1, 6))
C = FiniteSet(*range(2, 7))
for X0 in [X, A, B, C]:
    pprint(X0)

XS = [(A - (B | C), (A - B) & (A - C)),
      (A - (B & C), (A - B) | (A - C)),
      ((A | B) - C, (A - C) | (B - C)),
      ((A & B) - C, (A - C) & (B - C)),
      (A & (B - C), (A & B) - (A & C))]

for i, (L, R) in enumerate(XS):
    print(f'({chr(ord("a") + i)})')
    for X in [L, R]:
        pprint(X)
    print(L == R)
    print()

venn3_unweighted(subsets=(A, B, C))
plt.savefig('sample4.svg')

入出力結果(Terminal, Jupyter(IPython))

$ ./sample4.py
4.
{0, 1, 2, 3, 4, 5, 6}
{0, 1, 2, 3, 4}
{1, 2, 3, 4, 5}
{2, 3, 4, 5, 6}
(a)
{0}
{0}
True

(b)
{0, 1}
{0, 1}
True

(c)
{0, 1}
{0, 1}
True

(d)
{1}
{1}
True

(e)
{1}
{1}
True

$