数学 - Python - 集合と位相 - 集合と写像 - 集合間の演算(包含、和集合、共通部分、対称差(symmetric difference)、差)

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

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)、問題6、7を取り組んでみる。

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

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('6.')

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

print(A.is_subset(C))
print(A | (B & C) == (A | B) & C)

venn3_unweighted(subsets=(A, B, C))
plt.savefig('sample6.svg')

入出力結果(Terminal, Jupyter(IPython))

$ ./sample6.py
6.
{0, 1, 2, 3, 4, 5, 6}
{0, 1, 2, 3, 4}
{0, 1, 2, 3, 4, 5}
{0, 1, 2, 3, 4, 5, 6}
True
True
$

Kamimura's blog

2017年9月24日日曜日

数学 - Python - 集合と位相 - 集合と写像 - 集合間の演算(包含、和集合、共通部分、対称差(symmetric difference)、差)

0 コメント:

コメントを投稿