重溫離散系列②之良序原理

<div class="output_wrapper" id="output_wrapper_id" style="font-size: 15px; color: rgb(62, 62, 62); line-height: 1.8; word-spacing: 2px; letter-spacing: 2px; font-family: 'Helvetica Neue', Helvetica, 'Hiragino Sans GB', 'Microsoft YaHei', Arial, sans-serif; background-image: linear-gradient(90deg, rgba(50, 0, 0, 0.05) 3%, rgba(0, 0, 0, 0) 3%), linear-gradient(360deg, rgba(50, 0, 0, 0.05) 3%, rgba(0, 0, 0, 0) 3%); background-size: 20px 20px; background-position: center center;"><blockquote style="line-height: inherit; display: block; padding: 15px 15px 15px 1rem; font-size: 0.9em; margin: 1em 0px; color: rgb(0, 0, 0); border-left: 5px solid rgb(239, 112, 96); background: rgb(239, 235, 233); overflow: auto; overflow-wrap: normal; word-break: normal;"> <p style="font-size: inherit; color: inherit; line-height: inherit; padding: 0px; margin: 0px;">參考教材：<a href="https://book.douban.com/subject/33396340/" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; text-decoration: none; color: rgb(30, 107, 184); overflow-wrap: break-word;">計算機科學中的數學</a></p> <p style="font-size: inherit; color: inherit; line-height: inherit; padding: 0px; margin: 0px;">個人另外一篇博文：<a href="https://www.cnblogs.com/sang-bit/p/11778670.html" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; text-decoration: none; color: rgb(30, 107, 184); overflow-wrap: break-word;">重溫離散系列①之什麼是證實</a></p> </blockquote> <h3 id="h" style="color: inherit; line-height: inherit; padding: 0px; margin: 1.6em 0px; font-weight: bold; border-bottom: 2px solid rgb(239, 112, 96); font-size: 1.3em;"><span style="font-size: inherit; line-height: inherit; margin: 0px; display: inline-block; font-weight: normal; background: rgb(239, 112, 96); color: rgb(255, 255, 255); padding: 3px 10px 1px; border-top-right-radius: 3px; border-top-left-radius: 3px; margin-right: 3px;">良序原理</span><span style="display: inline-block; vertical-align: bottom; border-bottom: 36px solid rgb(239, 235, 233); border-right: 20px solid transparent;"> </span></h3> <blockquote style="line-height: inherit; display: block; padding: 15px 15px 15px 1rem; font-size: 0.9em; margin: 1em 0px; color: rgb(0, 0, 0); border-left: 5px solid rgb(239, 112, 96); background: rgb(239, 235, 233); overflow: auto; overflow-wrap: normal; word-break: normal;"> <p style="font-size: inherit; color: inherit; line-height: inherit; padding: 0px; margin: 0px;"><strong style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; font-weight: bold; color: rgb(233, 105, 0);">Definition：非空非負的整數集合必有最小元素。</strong></p> </blockquote> <p style="font-size: inherit; color: inherit; line-height: inherit; padding: 0px; margin: 1.7em 0px;">是的，你沒有看錯，良序原理就是這麼顯而易見。可是，良序原理倒是離散數學中最重要的原理之一。</p> <h3 id="h-1" style="color: inherit; line-height: inherit; padding: 0px; margin: 1.6em 0px; font-weight: bold; border-bottom: 2px solid rgb(239, 112, 96); font-size: 1.3em;"><span style="font-size: inherit; line-height: inherit; margin: 0px; display: inline-block; font-weight: normal; background: rgb(239, 112, 96); color: rgb(255, 255, 255); padding: 3px 10px 1px; border-top-right-radius: 3px; border-top-left-radius: 3px; margin-right: 3px;">良序證實</span><span style="display: inline-block; vertical-align: bottom; border-bottom: 36px solid rgb(239, 235, 233); border-right: 20px solid transparent;"> </span></h3> <blockquote style="line-height: inherit; display: block; padding: 15px 15px 15px 1rem; font-size: 0.9em; margin: 1em 0px; color: rgb(0, 0, 0); border-left: 5px solid rgb(239, 112, 96); background: rgb(239, 235, 233); overflow: auto; overflow-wrap: normal; word-break: normal;"> <p style="font-size: inherit; color: inherit; line-height: inherit; padding: 0px; margin: 0px;">良序證實是運用良序原理的一種證實方法。良序證實和反證法是掛鉤的，若是用到良序證實，就必定會用到反證法。</p> </blockquote> <p style="font-size: inherit; color: inherit; line-height: inherit; padding: 0px; margin: 1.7em 0px;">​ 咱們先看一道例題：</p> <p style="font-size: inherit; color: inherit; line-height: inherit; padding: 0px; margin: 1.7em 0px;">​ <strong style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; font-weight: bold; color: rgb(233, 105, 0);">例：證實對任意非負整數n，1+2+3+…..+n=n(n+1)/2</strong><br> </p><div align="center" style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px;"><img width="600" height="400" src="https://ae01.alicdn.com/kf/Hd8c8c7c21c064aec89222a94518149aa9.png" style="font-size: inherit; color: inherit; line-height: inherit; padding: 0px; display: block; margin: 0px auto; max-width: 100%;"></div><p style="font-size: inherit; color: inherit; line-height: inherit; padding: 0px; margin: 1.7em 0px;"></p> <p style="font-size: inherit; color: inherit; line-height: inherit; padding: 0px; margin: 1.7em 0px;">經過這道例題，我想你能基本感覺到良序定理的做用。咱們接着往下看：</p> <h4 id="h-2" style="color: inherit; line-height: inherit; padding: 0px; margin: 1.6em 0px; font-weight: bold; font-size: 1.2em;"><span style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px;">良序證實的模板</span></h4> <p style="font-size: inherit; color: inherit; line-height: inherit; padding: 0px; margin: 1.7em 0px;">使用良序定理證實"對全部n<span class="katex" style="font: 1.21em/1.2 KaTeX_Main, 'Times New Roman', serif; text-indent: 0px; text-rendering: auto; font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 8px 3px;"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABcAAAAqCAYAAABYzsDTAAABJUlEQVRYR+2Wva5EQBzFD0JsotBpNEQrW3sHtUQnEU/HExB0Pgq1l9Co0HAzm+ytzF3s3WSL+VeKcZz5nWPgtm3b8KHhmPgeWYZlt28MC8Ny/CBibfnCtgzDgCzL0LYt5nn+dXi/3xGGISRJoiZMDXRdV+R5jrquEQQBdF0Hz/PHqwKAKp6mKaZpguu6EAThlOhz8a44QZEkycOxLMuXhMlNu+IEBRnHcS4LU8UJ62VZoGkaVVxRFFiW9SeyXedHxFVVhWma4DjuXFs+ioUEGsfxI9Db7XaZO7WKXdeh73v4vv+/VSRWyV9eWZaoqgqe58EwDIiieGoXL4/ccRxRFAWapgG5fo5t24ii6Nrrf8oiZfFL5+88hIl/4ZeIBfoOARbocXo/lF77g5GDGowAAAAASUVORK5CYII=" style="font-size: inherit; color: inherit; line-height: inherit; padding: 0px; margin: 0px auto; max-width: 100%; display: inline-block; vertical-align: middle;"></span>N，p(n)成立。"（良序證實通常用於證實諸如此類問題</p> <ul style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px; padding-left: 32px; list-style-type: disc;"> <li style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px; margin-bottom: 0.5em;"><span style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px;">使用反證法，定義集合C爲P爲真的反例集合</span></li> <li style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px; margin-bottom: 0.5em;">根據良序原理，必定存在一個最小元素n<span class="katex" style="font: 1.21em/1.2 KaTeX_Main, 'Times New Roman', serif; text-indent: 0px; text-rendering: auto; font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 8px 3px;"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABcAAAAqCAYAAABYzsDTAAABJUlEQVRYR+2Wva5EQBzFD0JsotBpNEQrW3sHtUQnEU/HExB0Pgq1l9Co0HAzm+ytzF3s3WSL+VeKcZz5nWPgtm3b8KHhmPgeWYZlt28MC8Ny/CBibfnCtgzDgCzL0LYt5nn+dXi/3xGGISRJoiZMDXRdV+R5jrquEQQBdF0Hz/PHqwKAKp6mKaZpguu6EAThlOhz8a44QZEkycOxLMuXhMlNu+IEBRnHcS4LU8UJ62VZoGkaVVxRFFiW9SeyXedHxFVVhWma4DjuXFs+ioUEGsfxI9Db7XaZO7WKXdeh73v4vv+/VSRWyV9eWZaoqgqe58EwDIiieGoXL4/ccRxRFAWapgG5fo5t24ii6Nrrf8oiZfFL5+88hIl/4ZeIBfoOARbocXo/lF77g5GDGowAAAAASUVORK5CYII=" style="font-size: inherit; color: inherit; line-height: inherit; padding: 0px; margin: 0px auto; max-width: 100%; display: inline-block; vertical-align: middle;"></span>C</li> <li style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px; margin-bottom: 0.5em;"><span style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px;">得出矛盾----一般是P(n)爲真或C中存在一個比n更小的元素。這部分取決於具體的證實任務。</span></li> <li style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px; margin-bottom: 0.5em;"><span style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px;">得出結論，C必定是空集，即不存在反例。</span></li> </ul> <h3 id="h-3" style="color: inherit; line-height: inherit; padding: 0px; margin: 1.6em 0px; font-weight: bold; border-bottom: 2px solid rgb(239, 112, 96); font-size: 1.3em;"><span style="font-size: inherit; line-height: inherit; margin: 0px; display: inline-block; font-weight: normal; background: rgb(239, 112, 96); color: rgb(255, 255, 255); padding: 3px 10px 1px; border-top-right-radius: 3px; border-top-left-radius: 3px; margin-right: 3px;">良序集合</span><span style="display: inline-block; vertical-align: bottom; border-bottom: 36px solid rgb(239, 235, 233); border-right: 20px solid transparent;"> </span></h3> <blockquote style="line-height: inherit; display: block; padding: 15px 15px 15px 1rem; font-size: 0.9em; margin: 1em 0px; color: rgb(0, 0, 0); border-left: 5px solid rgb(239, 112, 96); background: rgb(239, 235, 233); overflow: auto; overflow-wrap: normal; word-break: normal;"> <p style="font-size: inherit; color: inherit; line-height: inherit; padding: 0px; margin: 0px;">若是一個集合的任意非空子集都有一個最小元素，咱們稱這個集合是良序的。</p> <p style="font-size: inherit; color: inherit; line-height: inherit; padding: 0px; margin: 0px;">（這個不是很重要，咱們就不詳細展開</p> </blockquote> <h3 id="h-4" style="color: inherit; line-height: inherit; padding: 0px; margin: 1.6em 0px; font-weight: bold; border-bottom: 2px solid rgb(239, 112, 96); font-size: 1.3em;"><span style="font-size: inherit; line-height: inherit; margin: 0px; display: inline-block; font-weight: normal; background: rgb(239, 112, 96); color: rgb(255, 255, 255); padding: 3px 10px 1px; border-top-right-radius: 3px; border-top-left-radius: 3px; margin-right: 3px;">一些習題</span><span style="display: inline-block; vertical-align: bottom; border-bottom: 36px solid rgb(239, 235, 233); border-right: 20px solid transparent;"> </span></h3> <blockquote style="line-height: inherit; display: block; padding: 15px 15px 15px 1rem; font-size: 0.9em; margin: 1em 0px; color: rgb(0, 0, 0); border-left: 5px solid rgb(239, 112, 96); background: rgb(239, 235, 233); overflow: auto; overflow-wrap: normal; word-break: normal;"> <p style="font-size: inherit; color: inherit; line-height: inherit; padding: 0px; margin: 0px;">我的認爲要想深刻理解和使用良序證實，是須要多從習題中總結提煉的，如下是一些良序證實的習題：<br><a href="http://www.baige.me/v?i=SYJ" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; text-decoration: none; color: rgb(30, 107, 184); overflow-wrap: break-word;">一些習題</a></p> </blockquote> <h3 id="h-5" style="color: inherit; line-height: inherit; padding: 0px; margin: 1.6em 0px; font-weight: bold; border-bottom: 2px solid rgb(239, 112, 96); font-size: 1.3em;"><span style="font-size: inherit; line-height: inherit; margin: 0px; display: inline-block; font-weight: normal; background: rgb(239, 112, 96); color: rgb(255, 255, 255); padding: 3px 10px 1px; border-top-right-radius: 3px; border-top-left-radius: 3px; margin-right: 3px;">總結</span><span style="display: inline-block; vertical-align: bottom; border-bottom: 36px solid rgb(239, 235, 233); border-right: 20px solid transparent;"> </span></h3> <blockquote style="line-height: inherit; display: block; padding: 15px 15px 15px 1rem; font-size: 0.9em; margin: 1em 0px; color: rgb(0, 0, 0); border-left: 5px solid rgb(239, 112, 96); background: rgb(239, 235, 233); overflow: auto; overflow-wrap: normal; word-break: normal;"> <p style="font-size: inherit; color: inherit; line-height: inherit; padding: 0px; margin: 0px;">良序原理是「基本的思惟定理」，而良序證實是基於良序原理的一種數學證實方法。通常用於證實諸如" 對全部n<span class="katex" style="font: 1.21em/1.2 KaTeX_Main, 'Times New Roman', serif; text-indent: 0px; text-rendering: auto; font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 8px 3px;"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABYAAAAoCAYAAAD6xArmAAABCklEQVRIS+2VMQ5EQBiFn0Q0SlGKWuIEKo2GRKegUDmLRCEKtaNQuIBExwGUGhEVNjbZzSaLsLuyzUw9882f917eT83zPOOCQxHwQ1UixTNfRAoixXvZkFT8ORXLDqjrGmmaouu65zSyLMMwjN31sGneAoqiCIIgQNd1cBwHmqYP75pV8DAM8H0fjuNAkqTDsNeLq+AsyzCOIzRN+wi6PFoFJ0kC0zTB8/xvwUEQoKoqMAyzClYUBa7rnjfvsokv07jve8RxDNu2IYriRzpv5rhtW4RhiEVPVVXBsuypD3bbbZomFEWBPM9RliWaprnDLcuC53nnzTs12sZl0sd/7uNvTCTmEfPe83MDqnrqiQUF998AAAAASUVORK5CYII=" style="font-size: inherit; color: inherit; line-height: inherit; padding: 0px; margin: 0px auto; max-width: 100%; display: inline-block; vertical-align: middle;"></span>N，p(n)成立 "此類問題。</p> </blockquote></div>

原文出處：https://www.cnblogs.com/sang-bit/p/11795387.html