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
集合論入門(基礎数学シリーズ)(松村 英之(著)、朝倉書店)の1.(集合算)、1.8(関係、同値関係、商集合)の練習問題12.を取り組んでみる。
-
x1、x2を任意の実数とする。
kを整数とすると次のことが成り立つ。
よってfに対応するRの同値関係はの各同値類は、その元の差が整数となる。
コード(Emacs)
Python 3
#!/usr/bin/env python3
from sympy import pprint, symbols, sin, cos, pi, solve
x = symbols('x', real=True)
for n in range(10):
k = n - 5
pprint(2 * pi * (x + k))
print(cos(2 * pi * x) == cos(2 * pi * (x + k)).expand()
and sin(2 * pi * x) == sin(2 * pi * (x + k)).expand())
print()
入出力結果(Terminal, Jupyter(IPython))
$ ./sample12.py 2⋅π⋅(x - 5) True 2⋅π⋅(x - 4) True 2⋅π⋅(x - 3) True 2⋅π⋅(x - 2) True 2⋅π⋅(x - 1) True 2⋅π⋅x True 2⋅π⋅(x + 1) True 2⋅π⋅(x + 2) True 2⋅π⋅(x + 3) True 2⋅π⋅(x + 4) True $
HTML5
<div id="graph0"></div> <pre id="output0"></pre> <label for="r0">r = </label> <input id="r0" type="number" min="0" value="0.5"> <label for="dx">dx = </label> <input id="dx" type="number" min="0" step="0.0001" value="0.001"> <br> <label for="x1">x1 = </label> <input id="x1" type="number" value="-2"> <label for="x2">x2 = </label> <input id="x2" type="number" value="2"> <br> <label for="y1">y1 = </label> <input id="y1" type="number" value="-2"> <label for="y2">y2 = </label> <input id="y2" type="number" value="2"> <br> <label for="x3">x3 = </label> <input id="x3" type="number" step="0.01" value="1.2"> <label for="x4">x4 = </label> <input id="x4" type="number" step="0.01" value="2.2"> <button id="draw0">draw</button> <button id="clear0">clear</button> <script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/d3/4.2.6/d3.min.js" integrity="sha256-5idA201uSwHAROtCops7codXJ0vja+6wbBrZdQ6ETQc=" crossorigin="anonymous"></script> <script src="sample12.js"></script>
JavaScript
let div0 = document.querySelector('#graph0'),
pre0 = document.querySelector('#output0'),
width = 600,
height = 600,
padding = 50,
btn0 = document.querySelector('#draw0'),
btn1 = document.querySelector('#clear0'),
input_r = document.querySelector('#r0'),
input_dx = document.querySelector('#dx'),
input_x1 = document.querySelector('#x1'),
input_x2 = document.querySelector('#x2'),
input_y1 = document.querySelector('#y1'),
input_y2 = document.querySelector('#y2'),
input_x3 = document.querySelector('#x3'),
input_x4 = document.querySelector('#x4'),
inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
input_x3, input_x4],
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 f1 = (x) => Math.sqrt(1 - (x + 1) ** 2),
f2 = (x) => -Math.sqrt(1 - (x + 1) ** 2),
g1 = (x) => Math.sqrt(1 - (x - 1) ** 2),
g2 = (x) => -Math.sqrt(1 - (x - 1) ** 2);
let draw = () => {
pre0.textContent = '';
let r = parseFloat(input_r.value),
dx = parseFloat(input_dx.value),
x1 = parseFloat(input_x1.value),
x2 = parseFloat(input_x2.value),
y1 = parseFloat(input_y1.value),
y2 = parseFloat(input_y2.value),
x3 = parseFloat(input_x3.value),
x4 = parseFloat(input_x4.value);
if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
return;
}
let points = [],
lines = [[0 - 1, 0, Math.cos(2 * Math.PI * x3) - 1,
Math.sin(2 * Math.PI * x3), 'green'],
[0 + 1, 0, Math.cos(2 * Math.PI * x4) + 1,
Math.sin(2 * Math.PI * x4), 'brown'],],
fns = [[f1, 'red'],
[f2, 'red'],
[g1, 'blue'],
[g2, 'blue']],
fns1 = [],
fns2 = [];
fns
.forEach((o) => {
let [f, color] = o;
for (let x = x1; x <= x2; x += dx) {
let y = f(x);
points.push([x, y, color]);
}
});
fns2
.forEach((o) => {
let [f, color] = o;
for (let x = x1; x <= x2; x += dx0) {
let g = f(x);
lines.push([x1, g(x1), x2, g(x2), color]);
}
});
let xscale = d3.scaleLinear()
.domain([x1, x2])
.range([padding, width - padding]);
let yscale = d3.scaleLinear()
.domain([y1, y2])
.range([height - padding, padding]);
let xaxis = d3.axisBottom().scale(xscale);
let yaxis = d3.axisLeft().scale(yscale);
div0.innerHTML = '';
let svg = d3.select('#graph0')
.append('svg')
.attr('width', width)
.attr('height', height);
svg.selectAll('line')
.data([[x1, 0, x2, 0], [0, y1, 0, y2]].concat(lines))
.enter()
.append('line')
.attr('x1', (d) => xscale(d[0]))
.attr('y1', (d) => yscale(d[1]))
.attr('x2', (d) => xscale(d[2]))
.attr('y2', (d) => yscale(d[3]))
.attr('stroke', (d) => d[4] || 'black');
svg.selectAll('circle')
.data(points)
.enter()
.append('circle')
.attr('cx', (d) => xscale(d[0]))
.attr('cy', (d) => yscale(d[1]))
.attr('r', r)
.attr('fill', (d) => d[2] || 'green');
svg.append('g')
.attr('transform', `translate(0, ${height - padding})`)
.call(xaxis);
svg.append('g')
.attr('transform', `translate(${padding}, 0)`)
.call(yaxis);
[fns, fns1, fns2].forEach((fs) => p(fs.join('\n')));
p(`x3 - x4 = ${x3 - x4}`);
};
inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();
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