数学 – Python - 集合と写像 - 集合の間の演算(対称差(symmetric difference))

集合・位相入門

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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(集合の間の演算)、問題7.を取り組んでみる。

問題7.

$(A - B) \cup (B - A) = (B - A) \cup (A - B)$
$\begin{array}{l} x \in (A \cup B) - (A \cap B) \\ \Leftrightarrow x \in (A \cup B) \land x \in {(A \cap B)}^{c} \\ \Leftrightarrow (x \in A \lor x \in B) \land x \in (A^{c} \cup B^{c}) \\ \Leftrightarrow (x \in A \lor x \in B) \land (x \in A^{c} \lor x \in B^{c}) \\ \Leftrightarrow (x \in A^{c} \land x \in B) \lor (x \in A \land x \in B^{c}) \\ \Leftrightarrow x \in (A^{c} \cap B) \cup (A \cap B^{c}) \end{array}$
$\begin{array}{l} ((A \cap B^{c}) \cup (A^{c} \cap B)) Δ C \\ = (((A \cap B^{c}) \cup (A^{c} \cap B)) \cap C^{c}) \cup ({((A \cap B^{c}) \cup (A^{c} \cap B))}^{c} \cap C) \\ = ((A \cap B^{c} \cap C^{c}) \cup (A^{c} \cap B \cap C^{c})) \cup (({(A \cap B^{c})}^{c} \cap {(A^{c} \cap B)}^{c}) \cap C) \\ = ((A \cap B^{c} \cap C^{c}) \cup (A^{c} \cap B \cap C^{c})) \cup (((A^{c} \cup B) \cap (A \cup B^{c})) \cap C) \\ = ((A \cap B^{c} \cap C^{c}) \cup (A^{c} \cap B \cap C^{c})) \cup (((A^{c} \cap B^{c}) \cup (A \cap B)) \cap C) \\ = ((A \cap B^{c} \cap C^{c}) \cup (A^{c} \cap B \cap C^{c})) \cup ((A^{c} \cap B^{c} \cap C) \cup (A \cap B \cap C)) \\ A Δ ((B \cap C^{c}) \cup (B^{c} \cap C)) \\ = (A \cap {((B \cap C^{c}) \cup (B^{c} \cap C))}^{c}) \cup (A^{c} \cap ((B \cap C^{c}) \cup (B^{c} \cap C))) \\ = (A \cap ((B^{c} \cup C) \cap (B \cup C^{c}))) \cup ((A^{c} \cap B \cap C^{c}) \cup (A^{c} \cap B^{c} \cap C)) \\ = (A \cap ((B^{c} \cap C^{c}) \cup (C \cap B))) \cup ((A^{c} \cap B \cap C^{c}) \cup (A^{c} \cap B^{c} \cap C)) \\ = ((A \cap B^{c} \cap C^{c}) \cup (A \cap B \cap C)) \cup ((A^{c} \cap B \cap C^{c}) \cup (A^{c} \cap B^{c} \cap C)) \end{array}$
$\begin{array}{l} A \cap ((B \cap C^{c}) \cup (B^{c} \cap C)) \\ = (A \cap B \cap C^{c}) \cup (A \cap B^{c} \cap C) \\ ((A \cap B) \cap {(A \cap C)}^{c}) \cup ({(A \cap B)}^{c} \cap (A \cap C)) \\ = ((A \cap B) \cap (A^{c} \cup C^{c})) \cup ((A^{c} \cup B^{c}) \cap (A \cap C)) \\ = (A \cap B \cap C^{c}) \cup (A \cap B^{c} \cap C) \end{array}$

コード(Emacs)

python 3.5

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

from matplotlib_venn import venn3_unweighted
import matplotlib.pyplot as plt
import sympy

a = sympy.FiniteSet(*range(1, 15))
b = sympy.FiniteSet(*range(5, 20))
c = sympy.FiniteSet(*range(10, 25))

expr = [
    (a.symmetric_difference(b),
     b.symmetric_difference(a)),
    (a.symmetric_difference(b),
     (a | b) - (a & b)),
    ((a.symmetric_difference(b)).symmetric_difference(c),
     (a.symmetric_difference(b.symmetric_difference(c)))),
    (a & b.symmetric_difference(c),
     (a & b).symmetric_difference(a & c))
]

for i, (left, right) in enumerate(expr):
    print('{0}: {1}'.format(chr(ord('a') + i), left == right))

plt.figure(figsize=(6, 6))
venn3_unweighted(subsets=(a, b, c), set_labels=('A', 'B', 'C'))
plt.savefig('sample7.svg')
plt.show()

入出力結果(Terminal, IPython)

$ ./sample7.py
a: True
b: True
c: True
d: True
$

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2016年10月2日日曜日

数学 – Python - 集合と写像 - 集合の間の演算(対称差(symmetric difference))

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