重温离散系列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;">参考教材:<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;">计算机科学中的数学</a></p> <p style="font-size: inherit; color: inherit; line-height: inherit; padding: 0px; margin: 0px;">个人另外一篇博文:<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;">重温离散系列①之什么是证实</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:非空非负的整数集合必有最小元素。</strong></p> </blockquote> <p style="font-size: inherit; color: inherit; line-height: inherit; padding: 0px; margin: 1.7em 0px;">是的,你没有看错,良序原理就是这么显而易见。可是,良序原理倒是离散数学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;">良序证实</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;">良序证实是运用良序原理的一种证实方法。良序证实和反证法是挂钩的,若是用到良序证实,就必定会用到反证法。</p> </blockquote> <p style="font-size: inherit; color: inherit; line-height: inherit; padding: 0px; margin: 1.7em 0px;">​ 咱们先看一道例题:</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:证实对任意非负整数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;">经过这道例题,我想你能基本感觉到良序定理的做用。咱们接着往下看:</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;">良序证实的模板</span></h4> <p style="font-size: inherit; color: inherit; line-height: inherit; padding: 0px; margin: 1.7em 0px;">使用良序定理证实"对全部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。"(良序证实通常用于证实诸如此类问题</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;">使用反证法,定义集合C为P为真的反例集合</span></li> <li style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px; margin-bottom: 0.5em;">根据良序原理,必定存在一个最小元素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)为真或C中存在一个比n更小的元素。这部分取决于具体的证实任务。</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;">得出结论,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;">若是一个集合的任意非空子集都有一个最小元素,咱们称这个集合是良序的。</p> <p style="font-size: inherit; color: inherit; line-height: inherit; padding: 0px; margin: 0px;">(这个不是很重要,咱们就不详细展开</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;">一些习题</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;">我的认为要想深刻理解和使用良序证实,是须要多从习题中总结提炼的,如下是一些良序证实的习题:<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;">一些习题</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;">总结</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”,而良序证实是基于良序原理的一种数学证实方法。通常用于证实诸如" 对全部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 "此类问题。</p> </blockquote></div>html

原文出处:https://www.cnblogs.com/sang-bit/p/11795387.htmlapp

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