2020年5月16日土曜日

学習環境

解析入門(中) (松坂和夫 数学入門シリーズ 5) (松坂 和夫(著)、岩波書店)の第11章(集合論初歩)、11.2(濃度)、問題9の解答を求めてみる。


  1. g x = x x + 1 g x x + 1 = x x g x - 1 = - g x x = g x 1 - g x X 0 = 1 f 1 = 1 g f 1 = 1 2 g f 1 2 = 1 2 1 2 + 1 = 1 1 + 2 = 1 3 g f 1 k = 1 k 1 k + 1 = 1 k + 1

    よって、 求める全単射 F は

    F : I J F : 0 , 1 0 , F x = { f x = x x = 1 k g - 1 x = x 1 - x x 1 k k - 0

コード

#!/usr/bin/env python3
from sympy import Rational
import matplotlib.pyplot as plt

print('9.')

n = 5


def frange(start, stop, step=0.05):
    result = []
    while start < stop:
        result.append(start)
        start += step
    return result


def f(x):
    if x in [Rational(1, k) for k in range(1, n + 1)]:
        return x
    return x / (1 - x)


xs = frange(0, Rational(1, n)) + [Rational(1, n)]
for k in range(n - 1):
    xs += frange(Rational(1, 5 - k) + 0.001,
                 Rational(1, 5 - k - 1)) + \
        [Rational(1, 5 - k - 1)]

ys = [f(x) for x in xs]
for x, y in zip(xs, ys):
    print((x, y))

plt.plot(xs,  ys)
plt.savefig('sample9.png')

入出力結果(Zsh、PowerShell、Terminal、Jupyter(IPython))

% ./sample9.py
9.
(0, 0.0)
(0.05, 0.052631578947368425)
(0.1, 0.11111111111111112)
(0.15000000000000002, 0.17647058823529416)
(1/5, 1/5)
(0.201000000000000, 0.251564455569462)
(1/4, 1/4)
(0.251000000000000, 0.335113484646195)
(0.301000000000000, 0.430615164520744)
(1/3, 1/3)
(0.334333333333333, 0.502253380070105)
(0.384333333333333, 0.624255549539794)
(0.434333333333333, 0.767825574543312)
(0.484333333333333, 0.939237233354880)
(1/2, 1/2)
(0.501000000000000, 1.00400801603206)
(0.551000000000000, 1.22717149220490)
(0.601000000000000, 1.50626566416040)
(0.651000000000000, 1.86532951289398)
(0.701000000000000, 2.34448160535117)
(0.751000000000000, 3.01606425702812)
(0.801000000000000, 4.02512562814071)
(0.851000000000000, 5.71140939597317)
(0.901000000000000, 9.10101010101014)
(0.951000000000000, 19.4081632653063)
(1, 1)
%

0 コメント:

コメントを投稿