数学 - Python - JavaScript - 無限をかぞえる - 集合論へのプレリュード – 可算集合 - 可算集合の性質(2)(可算集合の直積)

数学読本〈6〉

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)
• Nebo(Windows アプリ)
• iPad Pro + Apple Pencil
• MyScript Nebo(iPad アプリ)
• 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

数学読本〈6〉線形写像・1次変換/数論へのプレリュード/集合論へのプレリュード/εとδ/落ち穂拾い など(松坂 和夫(著)、岩波書店)の第24章(無限をかぞえる - 集合論へのプレリュード)、24.2(可算集合)、可算集合の性質(2)、問1.を取り組んでみる。

1. 
   
   $\begin{array}{}g\left(m,n\right)\\ =\left(1+2+\dots \; <!--\; horizontal\; ellipsis\; -->+\left(m+\left(n-1\right)-1\right)\right)+n\\ =\left(1+2+\dots \; <!--\; horizontal\; ellipsis\; -->+\left(m+n-2\right)\right)+n\\ =\frac{\left(m+n-2\right)\left(m+n-2+1\right)}{2}+n\\ =\frac{\left(m+n-1\right)\left(m+n-2\right)}{2}+n\end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3

from sympy import pprint, Rational


def g(m, n):
    return Rational((m + n - 1) * (m + n - 2), 2) + n

d = {}
for m in range(1, 10):
    for n in range(1, m + 1):
        d[g(m, n)] = f'g({m}, {n})'
        m -= 1

for n in sorted(d):
    print(f'{d[n]} = {n}')

入出力結果(Terminal, Jupyter(IPython))

$ ./sample1.py
g(1, 1) = 1
g(2, 1) = 2
g(1, 2) = 3
g(3, 1) = 4
g(2, 2) = 5
g(1, 3) = 6
g(4, 1) = 7
g(3, 2) = 8
g(2, 3) = 9
g(1, 4) = 10
g(5, 1) = 11
g(4, 2) = 12
g(3, 3) = 13
g(2, 4) = 14
g(1, 5) = 15
g(6, 1) = 16
g(5, 2) = 17
g(4, 3) = 18
g(3, 4) = 19
g(2, 5) = 20
g(1, 6) = 21
g(7, 1) = 22
g(6, 2) = 23
g(5, 3) = 24
g(4, 4) = 25
g(3, 5) = 26
g(2, 6) = 27
g(1, 7) = 28
g(8, 1) = 29
g(7, 2) = 30
g(6, 3) = 31
g(5, 4) = 32
g(4, 5) = 33
g(3, 6) = 34
g(2, 7) = 35
g(1, 8) = 36
g(9, 1) = 37
g(8, 2) = 38
g(7, 3) = 39
g(6, 4) = 40
g(5, 5) = 41
g(4, 6) = 42
g(3, 7) = 43
g(2, 8) = 44
g(1, 9) = 45
$

HTML5

<pre id="output0"></pre>
<label for="m0">m = </label>
<input id="m0" type="number" min="1" step="1" value="1">
<label for="n0">n = </label>
<input id="n0" type="number" min="1" step="1" value="1">

<button id="run0">run</button>
<button id="clear0">clear</button>

<script src="sample1.js"></script>

JavaScript

let pre0 = document.querySelector('#output0'),
    btn0 = document.querySelector('#run0'),
    btn1 = document.querySelector('#clear0'),
    input_m = document.querySelector('#m0'),
    input_n = document.querySelector('#n0'),
    inputs = [input_m, input_n],
    p = (x) => pre0.textContent += x + '\n',
    range = (start, end, step=1) => {
        let res = [];
        for (let i = start; i < end; i += step) {
            res.push(i);
        }
        return res;
    };

let g = (m, n) => (m + n - 1) * (m + n - 2) / 2 + n;

let output = () => {
    let m = parseInt(input_m.value, 10),
        n = parseInt(input_n.value, 10);

    p(`g(${m}, ${n}) = ${g(m, n)}`);
};

inputs.forEach((input) => input.onchange = output);
btn0.onclick = output;
btn1.onclick = () => pre0.textContent = '';
output();

m = n =