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Signed-off-by: Abdulkadir Furkan Şanlı <me@abdulocra.cy>
This commit is contained in:
Abdulkadir Furkan Şanlı 2020-12-26 00:42:59 +01:00
parent a2238e26f1
commit 441057ecee
Signed by: afk
GPG Key ID: C8F00588EE6ED1FE
35 changed files with 1543 additions and 559 deletions
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27
config.toml
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@ -1,36 +1,31 @@
baseURL = "https://abdulocra.cy/" baseURL = "https://abdulocra.cy/"
languageCode = "en-us" languageCode = "en-us"
title = "abdulocracy"
baseurl = "/"
theme = "terminal" theme = "terminal"
paginate = 5 paginate = 5
[params] [params]
# dir name of your blog content (default is `content/posts`) contentTypeName = "blog"
contentTypeName = "posts" themeColor = "green"
# ["orange", "blue", "red", "green", "pink"]
themeColor = "pink"
# if you set this to 0, only submenu trigger will be visible
showMenuItems = 0 showMenuItems = 0
# show selector to switch language
showLanguageSelector = false showLanguageSelector = false
# set theme to full screen width
fullWidthTheme = false fullWidthTheme = false
# center theme with default width
centerTheme = true centerTheme = true
# set a custom favicon (default is a `themeColor` square) favicon = "favicon.png"
favicon = "img/favicon/favicon.png" enableGitInfo = true
showLastUpdated = true
updatedDatePrefix = "updated"
[languages] [languages]
[languages.en] [languages.en]
languageName = "English" languageName = "English"
title = "abdulocracy's personal site" title = "abdulocracy's personal site"
owner = "Abdulkadir Furkan Şanlı"
subtitle = "" subtitle = ""
keywords = "" keywords = ""
copyright = "© Abdulkadir Furkan Şanlı 2019" copyright = ""
menuMore = "menu" menuMore = "menu"
readMore = "read more" readMore = "more"
readOtherPosts = "read other posts" readOtherPosts = "other posts"
[languages.en.params.logo] [languages.en.params.logo]
logoText = "abdulocracy" logoText = "abdulocracy"
@ -44,7 +39,7 @@ paginate = 5
[[languages.en.menu.main]] [[languages.en.menu.main]]
identifier = "uni-notes" identifier = "uni-notes"
name = "university notes" name = "university notes"
url = "/tags/university-notes" url = "/university-notes"
[[languages.en.menu.main]] [[languages.en.menu.main]]
identifier = "music" identifier = "music"
name = "music" name = "music"

2
content/about.md
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@ -1,6 +1,6 @@
--- ---
title: "about" title: "about"
date: 2020-06-10 date: 2019-11-04
--- ---
<image src="face.jpg" width="173" height="150" /> <image src="face.jpg" width="173" height="150" />

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content/blog/0.md Normal file
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@ -0,0 +1,10 @@
---
title: 0
date: "2020-12-25"
---
You found my site. Congratulations.
If there's content on here, how wonderful. If not, then I haven't yet realized my vague plans for a blog.
Fare thee well.

0
content/img/favicon/favicon.png → content/favicon.png
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11
content/posts/eidma.md → content/university-notes/eidma.md
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@ -1,9 +1,8 @@
+++ ---
title = "Introduction to Discrete Mathematics" title: Introduction to Discrete Mathematics
date = "2019-11-20" date: "2019-11-20"
tags = ["university-notes"] markup: pandoc
markup = "pandoc" ---
+++
- Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets. - Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets.

18
layouts/partials/extended_footer.html
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@ -1,8 +1,3 @@
<!--
To add an extended footer section, please create
`layouts/partials/extended_footer.html` in your Hugo directory.
-->
<!-- KaTeX --> <!-- KaTeX -->
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.11.1/dist/katex.min.css" <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.11.1/dist/katex.min.css"
integrity="sha384-zB1R0rpPzHqg7Kpt0Aljp8JPLqbXI3bhnPWROx27a9N0Ll6ZP/+DiW/UqRcLbRjq" crossorigin="anonymous"> integrity="sha384-zB1R0rpPzHqg7Kpt0Aljp8JPLqbXI3bhnPWROx27a9N0Ll6ZP/+DiW/UqRcLbRjq" crossorigin="anonymous">
@ -12,16 +7,3 @@ To add an extended footer section, please create
<script defer src="https://cdn.jsdelivr.net/npm/katex@0.11.1/dist/contrib/auto-render.min.js" <script defer src="https://cdn.jsdelivr.net/npm/katex@0.11.1/dist/contrib/auto-render.min.js"
integrity="sha384-kWPLUVMOks5AQFrykwIup5lo0m3iMkkHrD0uJ4H5cjeGihAutqP0yW0J6dpFiVkI" crossorigin="anonymous" integrity="sha384-kWPLUVMOks5AQFrykwIup5lo0m3iMkkHrD0uJ4H5cjeGihAutqP0yW0J6dpFiVkI" crossorigin="anonymous"
onload="renderMathInElement(document.body);"></script> onload="renderMathInElement(document.body);"></script>
<!-- MathJax
<script>
MathJax = {
tex: {
inlineMath: [['$', '$'], ['\\(', '\\)']],
displayMath: [['$$', '$$'], ['\[', '\]']]
}
};
</script>
<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js">
</script>
-->

19
layouts/partials/footer.html
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@ -1,20 +1,17 @@
<footer class="footer"> <footer class="footer">
<div class="footer__inner"> <div class="footer__inner">
{{ if $.Site.Copyright }}
<div class="copyright copyright--user"> <div class="copyright copyright--user">
<span>{{ $.Site.Copyright | safeHTML }} :: <a href="https://creativecommons.org/licenses/by-nd/4.0/">CC <span>© Abdulkadir Furkan Şanlı {{ now.Year }} :: <a href="https://creativecommons.org/licenses/by-nd/4.0/">CC
BY-ND</a></span> BY-ND</a> :: theme by <a href="https://twitter.com/panr">panr</a></span>
{{else}}
<div class="copyright">
<span>© {{ now.Year }} Powered by <a href="http://gohugo.io">Hugo</a></span>
{{end}}
<span>:: theme by <a href="https://twitter.com/panr">panr</a></span>
</div>
</div> </div>
</footer> </footer>
<script src="{{ "assets/main.js" | absURL }}"></script> <script src="{{ " assets/main.js" | absURL }}"></script>
<script src="{{ "assets/prism.js" | absURL }}"></script> <script src="{{ " assets/prism.js" | absURL }}"></script>
{{ if $.Site.Params.showLanguageSelector }}
<script src="{{ " assets/languageSelector.js" | absURL }}"></script>
{{ end }}
<!-- Extended footer section--> <!-- Extended footer section-->
{{ partial "extended_footer.html" . }} {{ partial "extended_footer.html" . }}

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public/404.html Normal file
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@ -0,0 +1,176 @@
<!DOCTYPE html>
<html lang="en">
<head>
<title>404 Page not found :: abdulocracy&#39;s personal site</title>
<meta http-equiv="content-type" content="text/html; charset=utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<meta name="description" content="" />
<meta name="keywords" content="" />
<meta name="robots" content="noodp" />
<link rel="canonical" href="https://abdulocra.cy/404.html" />
<link rel="stylesheet" href="https://abdulocra.cy/assets/style.css">
<link rel="stylesheet" href="https://abdulocra.cy/assets/green.css">
<link rel="apple-touch-icon-precomposed" sizes="144x144" href="https://abdulocra.cy/img/apple-touch-icon-144-precomposed.png">
<link rel="shortcut icon" href="https://abdulocra.cy/favicon.png">
<meta name="twitter:card" content="summary" />
<meta property="og:locale" content="en" />
<meta property="og:type" content="website" />
<meta property="og:title" content="404 Page not found :: abdulocracy&#39;s personal site">
<meta property="og:description" content="" />
<meta property="og:url" content="https://abdulocra.cy/404.html" />
<meta property="og:site_name" content="404 Page not found" />
<meta property="og:image" content="https://abdulocra.cy/favicon.png">
<meta property="og:image:width" content="2048">
<meta property="og:image:height" content="1024">
</head>
<body class="green">
<div class="container center headings--one-size">
<header class="header">
<div class="header__inner">
<div class="header__logo">
<a href="/">
<div class="logo">
abdulocracy
</div>
</a>
</div>
<div class="menu-trigger">menu</div>
</div>
<nav class="menu">
<ul class="menu__inner menu__inner--desktop">
<ul class="menu__sub-inner">
<li class="menu__sub-inner-more-trigger">menu ▾</li>
<ul class="menu__sub-inner-more hidden">
<li><a href="/about">about</a></li>
<li><a href="https://music.abdulocra.cy">music</a></li>
<li><a href="/university-notes">university notes</a></li>
</ul>
</ul>
</ul>
<ul class="menu__inner menu__inner--mobile">
<li><a href="/about">about</a></li>
<li><a href="https://music.abdulocra.cy">music</a></li>
<li><a href="/university-notes">university notes</a></li>
</ul>
</nav>
</header>
<div class="content">
<div class="post">
<h1 class="post-title">404 — Page not found...</h1>
<div class="post-content">
<a href="https://abdulocra.cy/">Back to home page&nbsp;→</a>
</div>
</div>
</div>
<footer class="footer">
<div class="footer__inner">
<div class="copyright copyright--user">
<span>© Abdulkadir Furkan Şanlı 2020 :: <a href="https://creativecommons.org/licenses/by-nd/4.0/">CC
BY-ND</a> :: theme by <a href="https://twitter.com/panr">panr</a></span>
</div>
</footer>
<script src="https://abdulocra.cy/%20assets/main.js"></script>
<script src="https://abdulocra.cy/%20assets/prism.js"></script>
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.11.1/dist/katex.min.css"
integrity="sha384-zB1R0rpPzHqg7Kpt0Aljp8JPLqbXI3bhnPWROx27a9N0Ll6ZP/+DiW/UqRcLbRjq" crossorigin="anonymous">
<script defer src="https://cdn.jsdelivr.net/npm/katex@0.11.1/dist/katex.min.js"
integrity="sha384-y23I5Q6l+B6vatafAwxRu/0oK/79VlbSz7Q9aiSZUvyWYIYsd+qj+o24G5ZU2zJz"
crossorigin="anonymous"></script>
<script defer src="https://cdn.jsdelivr.net/npm/katex@0.11.1/dist/contrib/auto-render.min.js"
integrity="sha384-kWPLUVMOks5AQFrykwIup5lo0m3iMkkHrD0uJ4H5cjeGihAutqP0yW0J6dpFiVkI" crossorigin="anonymous"
onload="renderMathInElement(document.body);"></script>
</div>
</body>
</html>

71
public/about/index.html
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@ -5,16 +5,18 @@
<title>about :: abdulocracy&#39;s personal site</title> <title>about :: abdulocracy&#39;s personal site</title>
<meta http-equiv="content-type" content="text/html; charset=utf-8"> <meta http-equiv="content-type" content="text/html; charset=utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0, maximum-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1.0">
<meta name="description" content=" name: Abdulkadir Furkan Şanlı handle: abdulocracy contact: email: me at abdulocra dot cy gpg: 0xEE6ED1FE irc (freenode): abdulocracy "/> <meta name="description" content=" name: Abdulkadir Furkan Şanlı handle: abdulocracy contact: email: me at abdulocra dot cy gpg: 0xEE6ED1FE irc (freenode): abdulocracy " />
<meta name="keywords" content=""/> <meta name="keywords" content="" />
<meta name="robots" content="noodp"/> <meta name="robots" content="noodp" />
<link rel="canonical" href="https://abdulocra.cy/about/" /> <link rel="canonical" href="https://abdulocra.cy/about/" />
<link rel="stylesheet" href="https://abdulocra.cy/assets/style.css"> <link rel="stylesheet" href="https://abdulocra.cy/assets/style.css">
<link rel="stylesheet" href="https://abdulocra.cy/assets/pink.css"> <link rel="stylesheet" href="https://abdulocra.cy/assets/green.css">
@ -23,29 +25,31 @@
<link rel="apple-touch-icon-precomposed" sizes="144x144" href="https://abdulocra.cy/img/apple-touch-icon-144-precomposed.png"> <link rel="apple-touch-icon-precomposed" sizes="144x144" href="https://abdulocra.cy/img/apple-touch-icon-144-precomposed.png">
<link rel="shortcut icon" href="https://abdulocra.cy/img/favicon/favicon.png"> <link rel="shortcut icon" href="https://abdulocra.cy/favicon.png">
<meta name="twitter:card" content="summary" /> <meta name="twitter:card" content="summary" />
<meta name="twitter:title" content="about :: abdulocracy&#39;s personal site — " />
<meta name="twitter:description" content=" name: Abdulkadir Furkan Şanlı handle: abdulocracy contact: email: me at abdulocra dot cy gpg: 0xEE6ED1FE irc (freenode): abdulocracy " />
<meta name="twitter:site" content="https://abdulocra.cy/" />
<meta name="twitter:creator" content="" />
<meta name="twitter:image" content="">
<meta property="og:locale" content="en" /> <meta property="og:locale" content="en" />
<meta property="og:type" content="article" /> <meta property="og:type" content="article" />
<meta property="og:title" content="about :: abdulocracy&#39;s personal site — "> <meta property="og:title" content="about :: abdulocracy&#39;s personal site">
<meta property="og:description" content=" name: Abdulkadir Furkan Şanlı handle: abdulocracy contact: email: me at abdulocra dot cy gpg: 0xEE6ED1FE irc (freenode): abdulocracy " /> <meta property="og:description" content=" name: Abdulkadir Furkan Şanlı handle: abdulocracy contact: email: me at abdulocra dot cy gpg: 0xEE6ED1FE irc (freenode): abdulocracy " />
<meta property="og:url" content="https://abdulocra.cy/about/" /> <meta property="og:url" content="https://abdulocra.cy/about/" />
<meta property="og:site_name" content="about" /> <meta property="og:site_name" content="about" />
<meta property="og:image" content="">
<meta property="og:image" content="https://abdulocra.cy/favicon.png">
<meta property="og:image:width" content="2048"> <meta property="og:image:width" content="2048">
<meta property="og:image:height" content="1024"> <meta property="og:image:height" content="1024">
<meta property="article:published_time" content="2020-06-10 00:00:00 &#43;0000 UTC" />
<meta property="article:published_time" content="2019-11-04 00:00:00 &#43;0000 UTC" />
@ -58,10 +62,10 @@
</head> </head>
<body class=""> <body class="green">
<div class="container center"> <div class="container center headings--one-size">
<header class="header"> <header class="header">
<div class="header__inner"> <div class="header__inner">
@ -95,7 +99,7 @@
<li><a href="/tags/university-notes">university notes</a></li> <li><a href="/university-notes">university notes</a></li>
</ul> </ul>
@ -117,7 +121,7 @@
<li><a href="/tags/university-notes">university notes</a></li> <li><a href="/university-notes">university notes</a></li>
@ -134,10 +138,10 @@
<h1 class="post-title"> <h1 class="post-title">
<a href="https://abdulocra.cy/about/">about</a></h1> <a href="https://abdulocra.cy/about/">about</a></h1>
<div class="post-meta"> <div class="post-meta">
<span class="post-date"> <span class="post-date">
2020-06-10 2019-11-04 [updated: 2019-11-04]
</span> </span>
</div> </div>
@ -146,8 +150,10 @@
<div class="post-content">
<p><image src="face.jpg" width="173" height="150" /></p>
<div class="post-content"><div>
<p><image src="face.jpg" width="173" height="150" /></p>
<ul> <ul>
<li>name: Abdulkadir Furkan Şanlı</li> <li>name: Abdulkadir Furkan Şanlı</li>
<li>handle: abdulocracy</li> <li>handle: abdulocracy</li>
@ -160,7 +166,8 @@
</li> </li>
</ul> </ul>
</div> </div></div>
@ -172,18 +179,14 @@
<footer class="footer"> <footer class="footer">
<div class="footer__inner"> <div class="footer__inner">
<div class="copyright copyright--user"> <div class="copyright copyright--user">
<span>© Abdulkadir Furkan Şanlı 2019 :: <a href="https://creativecommons.org/licenses/by-nd/4.0/">CC <span>© Abdulkadir Furkan Şanlı 2020 :: <a href="https://creativecommons.org/licenses/by-nd/4.0/">CC
BY-ND</a></span> BY-ND</a> :: theme by <a href="https://twitter.com/panr">panr</a></span>
<span>:: theme by <a href="https://twitter.com/panr">panr</a></span>
</div>
</div> </div>
</footer> </footer>
<script src="https://abdulocra.cy/assets/main.js"></script> <script src="https://abdulocra.cy/%20assets/main.js"></script>
<script src="https://abdulocra.cy/assets/prism.js"></script> <script src="https://abdulocra.cy/%20assets/prism.js"></script>
@ -199,8 +202,6 @@
onload="renderMathInElement(document.body);"></script> onload="renderMathInElement(document.body);"></script>
</div> </div>

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public/assets/main.js
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@ -1,133 +1 @@
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<!DOCTYPE html>
<html lang="en">
<head>
<title>0 :: abdulocracy&#39;s personal site</title>
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<meta name="description" content="You found my site. Congratulations.
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Fare thee well." />
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<meta property="og:type" content="article" />
<meta property="og:title" content="0 :: abdulocracy&#39;s personal site">
<meta property="og:description" content="You found my site. Congratulations.
If there&amp;rsquo;s content on here, how wonderful. If not, then I haven&amp;rsquo;t yet realized my vague plans for a blog.
Fare thee well." />
<meta property="og:url" content="https://abdulocra.cy/blog/0/" />
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<body class="green">
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abdulocracy
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<ul class="menu__sub-inner">
<li class="menu__sub-inner-more-trigger">menu ▾</li>
<ul class="menu__sub-inner-more hidden">
<li><a href="/about">about</a></li>
<li><a href="https://music.abdulocra.cy">music</a></li>
<li><a href="/university-notes">university notes</a></li>
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<li><a href="/university-notes">university notes</a></li>
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<div class="content">
<div class="post">
<h1 class="post-title">
<a href="https://abdulocra.cy/blog/0/">0</a></h1>
<div class="post-meta">
<span class="post-date">
2020-12-25 [updated: 2020-12-25]
</span>
</div>
<div class="post-content"><div>
<p>You found my site. Congratulations.</p>
<p>If there&rsquo;s content on here, how wonderful. If not, then I haven&rsquo;t yet realized my vague plans for a blog.</p>
<p>Fare thee well.</p>
</div></div>
</div>
</div>
<footer class="footer">
<div class="footer__inner">
<div class="copyright copyright--user">
<span>© Abdulkadir Furkan Şanlı 2020 :: <a href="https://creativecommons.org/licenses/by-nd/4.0/">CC
BY-ND</a> :: theme by <a href="https://twitter.com/panr">panr</a></span>
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</div>
</body>
</html>

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<!DOCTYPE html>
<html lang="en">
<head>
<title>Blogs :: abdulocracy&#39;s personal site</title>
<meta http-equiv="content-type" content="text/html; charset=utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<meta name="description" content="" />
<meta name="keywords" content="" />
<meta name="robots" content="noodp" />
<link rel="canonical" href="https://abdulocra.cy/blog/" />
<link rel="stylesheet" href="https://abdulocra.cy/assets/style.css">
<link rel="stylesheet" href="https://abdulocra.cy/assets/green.css">
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<meta property="og:title" content="Blogs :: abdulocracy&#39;s personal site">
<meta property="og:description" content="" />
<meta property="og:url" content="https://abdulocra.cy/blog/" />
<meta property="og:site_name" content="Blogs" />
<meta property="og:image" content="https://abdulocra.cy/favicon.png">
<meta property="og:image:width" content="2048">
<meta property="og:image:height" content="1024">
<link href="/blog/index.xml" rel="alternate" type="application/rss+xml" title="abdulocracy&#39;s personal site" />
</head>
<body class="green">
<div class="container center headings--one-size">
<header class="header">
<div class="header__inner">
<div class="header__logo">
<a href="/">
<div class="logo">
abdulocracy
</div>
</a>
</div>
<div class="menu-trigger">menu</div>
</div>
<nav class="menu">
<ul class="menu__inner menu__inner--desktop">
<ul class="menu__sub-inner">
<li class="menu__sub-inner-more-trigger">menu ▾</li>
<ul class="menu__sub-inner-more hidden">
<li><a href="/about">about</a></li>
<li><a href="https://music.abdulocra.cy">music</a></li>
<li><a href="/university-notes">university notes</a></li>
</ul>
</ul>
</ul>
<ul class="menu__inner menu__inner--mobile">
<li><a href="/about">about</a></li>
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<li><a href="/university-notes">university notes</a></li>
</ul>
</nav>
</header>
<div class="content">
<div class="posts">
<div class="post on-list">
<h1 class="post-title">
<a href="https://abdulocra.cy/blog/0/">0</a>
</h1>
<div class="post-meta">
<span class="post-date">
2020-12-25
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<div class="post-content">
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<div>
<a class="read-more button"
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</div>
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<div class="pagination">
<div class="pagination__buttons">
</div>
</div>
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</div>
<footer class="footer">
<div class="footer__inner">
<div class="copyright copyright--user">
<span>© Abdulkadir Furkan Şanlı 2020 :: <a href="https://creativecommons.org/licenses/by-nd/4.0/">CC
BY-ND</a> :: theme by <a href="https://twitter.com/panr">panr</a></span>
</div>
</footer>
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<title>Categories :: abdulocracy&#39;s personal site</title> <title>Categories :: abdulocracy&#39;s personal site</title>
<meta http-equiv="content-type" content="text/html; charset=utf-8"> <meta http-equiv="content-type" content="text/html; charset=utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0, maximum-scale=1"> <meta name="viewport" content="width=device-width, initial-scale=1.0">
<meta name="description" content=""/> <meta name="description" content="" />
<meta name="keywords" content=""/> <meta name="keywords" content="" />
<meta name="robots" content="noodp"/> <meta name="robots" content="noodp" />
<link rel="canonical" href="https://abdulocra.cy/categories/" /> <link rel="canonical" href="https://abdulocra.cy/categories/" />
<link rel="stylesheet" href="https://abdulocra.cy/assets/style.css"> <link rel="stylesheet" href="https://abdulocra.cy/assets/style.css">
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<meta name="twitter:site" content="https://abdulocra.cy/" />
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<span>© Abdulkadir Furkan Şanlı 2019 :: <a href="https://creativecommons.org/licenses/by-nd/4.0/">CC <span>© Abdulkadir Furkan Şanlı 2020 :: <a href="https://creativecommons.org/licenses/by-nd/4.0/">CC
BY-ND</a></span> BY-ND</a> :: theme by <a href="https://twitter.com/panr">panr</a></span>
<span>:: theme by <a href="https://twitter.com/panr">panr</a></span>
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@ -5,11 +5,6 @@
<link>https://abdulocra.cy/categories/</link> <link>https://abdulocra.cy/categories/</link>
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Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets. Propositional calculus Comes from the linguistic concept that things can be either true or false.
We should avoid variables when forming statements, as they may change the logical value.
(2=7) statement (x=5) not a statement In logic we do not use the equals sign, we use the equivalence sign (\equiv).
Logical values (booleans) are denoted by either 0 or 1 (or t, f, etc.
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We should avoid variables when forming statements, as they may change the logical value. We should avoid variables when forming statements, as they may change the logical value.
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2019-11-20 [updated: 2019-11-20]
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<li>Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets.</li>
</ul>
<h2 id="propositional-calculus">Propositional calculus<a href="#propositional-calculus" class="hanchor" ariaLabel="Anchor">&#8983;</a> </h2>
<ul>
<li><p>Comes from the linguistic concept that things can be either true or false.</p></li>
<li><p>We should avoid variables when forming statements, as they may change the logical value.</p>
<ul>
<li><span class="math inline">\(2=7\)</span> statement</li>
<li><span class="math inline">\(x=5\)</span> not a statement</li>
</ul></li>
<li><p>In logic we do not use the equals sign, we use the equivalence sign <span class="math inline">\(\equiv\)</span>.</p></li>
<li><p>Logical values (booleans) are denoted by either 0 or 1 (or t, f, etc.).</p></li>
<li><p>When doing logic, we use propositional variables (e.g. p, q, r).</p>
<ul>
<li>Can be either <strong>true</strong> or <strong>false</strong>.</li>
</ul></li>
<li><p>The operations done on propositional variables are called propositional connectives.</p>
<ul>
<li>Conjunction: <span class="math inline">\(p \land q\)</span> is only true if both p and q are true <span class="math inline">\((0001)\)</span></li>
<li>Disjunction: <span class="math inline">\(p \lor q\)</span> is only false if both p and q are false <span class="math inline">\((0111)\)</span></li>
<li>Implication (material conditional): <span class="math inline">\(p \implies q\)</span> is false only if p is true and q is false (truth table <span class="math inline">\((1011)\)</span>)
<ul>
<li><span class="math inline">\(\equiv \neg p \lor q\)</span></li>
</ul></li>
</ul></li>
<li><p>Not necessarily connectives but unary operations:</p>
<ul>
<li>Negation: Denoted by ~, <span class="math inline">\(\neg\)</span> or NOT, negates the one input <span class="math inline">\((10)\)</span>.</li>
</ul></li>
<li><p>A (propositional) formula is a “properly constructed” logical expression.</p>
<ul>
<li>e.g. <span class="math inline">\(\neg[(p \lor q)] \land r\)</span></li>
<li><span class="math inline">\((p \land)\)</span> is not a formula, as <span class="math inline">\(\land\)</span> requires 2 variables.</li>
<li>Logical equivalence: <span class="math inline">\(\phi(p, q, k) \equiv \psi(p, q, k)\)</span>, logical value of <span class="math inline">\(\phi\)</span> is equal to logical value of <span class="math inline">\(\psi\)</span>.</li>
<li>Commutativity: <span class="math inline">\(p \land q \equiv q \land p\)</span></li>
<li>Associativity: <span class="math inline">\((p \land q) \land r \equiv p \land (q \land r)\)</span></li>
<li>Distributivity: <span class="math inline">\(p \land (q \lor r) \equiv (p \land q) \lor (p \land r)\)</span></li>
<li>Conjunctive normal form: every formula can be written as a conjunction of one or more disjunctions.
<ul>
<li><span class="math inline">\(\neg(B \lor C)\)</span> can be written as <span class="math inline">\(\neg B \land \neg C\)</span></li>
</ul></li>
</ul></li>
<li><p>Double negation law: <span class="math inline">\(\neg(\neg p) \equiv p\)</span></p></li>
<li><p>De Morgan’s laws: <span class="math inline">\(\neg(p \land q) \equiv \neg p \lor \neg q\)</span> and <span class="math inline">\(\neg(p \lor q) \equiv \neg p \land \neg q\)</span>.</p></li>
<li><p>If and only if (<em>iff</em>): <span class="math inline">\(p \iff p \equiv (p \implies q) \land (q \implies p)\)</span></p></li>
<li><p>Contraposition law:</p>
<ul>
<li><span class="math inline">\((p \implies q) \equiv (\neg q \implies \neg p)\)</span> prove by contraposition
<ul>
<li><span class="math inline">\((p \implies q) \equiv (\neg p \lor q)\)</span></li>
<li><span class="math inline">\((\neg q \implies \neg p) \equiv (\neg (\neg q) \lor (\neg p) \equiv (q \lor \neg p) \equiv (\neg p \lor q)\)</span></li>
</ul></li>
</ul></li>
<li><p>Contradiction law:</p>
<ul>
<li><span class="math inline">\(p \lor \neg p \equiv 1\)</span> and <span class="math inline">\(p \land \neg p \equiv 0\)</span></li>
</ul></li>
<li><p>Tautology: <span class="math inline">\(\phi (p, q, ... r)\)</span> is a tautology <em>iff</em> <span class="math inline">\(\phi \equiv 1\)</span></p></li>
</ul>
<h2 id="sets">Sets<a href="#sets" class="hanchor" ariaLabel="Anchor">&#8983;</a> </h2>
<ul>
<li>We will consider subsets of universal set <span class="math inline">\(\mathbb X\)</span>
<ul>
<li><span class="math inline">\(2^\mathbb X = \{ A : A \subseteq \mathbb X\}\)</span></li>
<li><span class="math inline">\(2^\mathbb X = P(\mathbb X)\)</span></li>
<li>All 2 object subsets of <span class="math inline">\(\mathbb X\)</span>: <span class="math inline">\(P_2(\mathbb X)\)</span></li>
</ul></li>
<li><span class="math inline">\(A \subset B \equiv\)</span> every element of A is an element of B <span class="math inline">\(\equiv \{x \in \mathbb X : x \in A \implies x \in B\}\)</span></li>
<li>Operations on sets:
<ul>
<li>Union - <span class="math inline">\(\cup\)</span> - <span class="math inline">\(A \cup B = \{ x \in \mathbb X : x \in A \lor x \in B \}\)</span></li>
<li>Intersection - <span class="math inline">\(\cap\)</span> - <span class="math inline">\(A \cap B = \{ x \in \mathbb X : x \in A \land x \in B \}\)</span></li>
<li>Complement - <span class="math inline">\(A&#39;\)</span> - <span class="math inline">\(A&#39; = \{ x \in \mathbb X : \neg (x \in A) \}\)</span>
<ul>
<li>If <span class="math inline">\(x = \{ 1 \}\)</span> then <span class="math inline">\(x&#39; = \emptyset\)</span></li>
</ul></li>
</ul></li>
<li>Equality of sets: <span class="math inline">\(A = B\)</span> iff <span class="math inline">\(x \in \mathbb X : (x \in A \iff x \in B)\)</span></li>
<li>Difference of sets:
<ul>
<li><span class="math inline">\(A \setminus B = \{ x \in \mathbb X : x \in A \land x \notin B \} = A \cap B&#39;\)</span></li>
<li>Symmetric difference: <span class="math inline">\(A \div B = (A \setminus B) \cup (B \setminus A)\)</span></li>
</ul></li>
<li>Laws of set algebra:
<ul>
<li><span class="math inline">\(A \cup B = B \cup A , A \cap B = B \cap A\)</span></li>
<li><span class="math inline">\((A \cup B) \cup C = A \cup (B \cup C), (A \cap B) \cap C = A \cap (B \cap C)\)</span></li>
<li><span class="math inline">\((A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\)</span> vice versa</li>
<li><span class="math inline">\(A \cap \emptyset, A \cap \mathbb X = A, A \cup \emptyset = A, A \cup \mathbb X = \mathbb X\)</span></li>
<li><span class="math inline">\((A \cup B)&#39; = A&#39; \cap B&#39;\)</span> vice versa</li>
<li><span class="math inline">\(A \cup A&#39; = \mathbb X, A \cap A&#39; = \emptyset\)</span></li>
</ul></li>
<li>Note: <span class="math inline">\(\{ \emptyset \} \neq \emptyset\)</span>, one is a set with one element, one is the empty set, no elements (<span class="math inline">\(\{ \}\)</span>)</li>
<li>Quip: <span class="math inline">\(\{ x \in \mathbb R : x^2 = -1\} = \emptyset\)</span></li>
</ul>
<h2 id="quantifiers">Quantifiers<a href="#quantifiers" class="hanchor" ariaLabel="Anchor">&#8983;</a> </h2>
<ul>
<li><span class="math inline">\(\phi\)</span> - prepositional function: yields only true or false value</li>
<li><span class="math inline">\(\forall\)</span> means “for all” and <span class="math inline">\(\exists\)</span> means “there exists”</li>
<li><span class="math inline">\(\forall\)</span>:
<ul>
<li>Shorthand for <span class="math inline">\(\land\)</span> e.g. <span class="math inline">\((\forall x \in \{ 1, 2, ... 10 \}) x &gt; 0 \equiv 1 &gt; 0 \land 2 &gt; 0 \land ... 10 &gt; 0\)</span></li>
</ul></li>
<li><span class="math inline">\(\exists\)</span>:
<ul>
<li>Shorthand for <span class="math inline">\(\lor\)</span> e.g. <span class="math inline">\((\exists x \in \{ 1, 2, ... 10 \}) x &gt; 5 \equiv 1 &gt; 5 \lor 2 &gt; 5 \lor ... 10 &gt; 5\)</span></li>
</ul></li>
<li><span class="math inline">\(\neg \forall \equiv \exists\)</span>, vice versa</li>
<li>With quantifiers we can write logical statements e.g.
<ul>
<li><span class="math inline">\((\forall x \in \mathbb{R}) (\forall y \in \mathbb{R}) x &gt; y\)</span> is a statement and is false</li>
<li><span class="math inline">\((\forall x) (\exists y) x &gt; y\)</span> is true</li>
<li>shortcut: <span class="math inline">\((\exists x, y) \equiv (\exists x) (\exists y)\)</span></li>
</ul></li>
<li>Quantifiers can be expressed in set language, sort of a definition in terms of sets:
<ul>
<li><span class="math inline">\((\forall x \in \mathbb{X}) (\phi(x)) \equiv \{ p \in \mathbb{X} : \phi(p) \} = \mathbb{X}\)</span></li>
<li><span class="math inline">\((\exists x \in \mathbb{X}) (\phi(x)) \equiv \{ q \in \mathbb{X} : \phi(q) \} \neq \emptyset\)</span></li>
<li><span class="math inline">\((\exists x \in \mathbb{X}) (\neg \phi(x)) \equiv \neg ( \{ p \in \mathbb{X} : \phi(p) \} = \mathbb{X} )\)</span></li>
</ul></li>
<li>Order of quantifiers matters.</li>
</ul>
<h2 id="relations">Relations<a href="#relations" class="hanchor" ariaLabel="Anchor">&#8983;</a> </h2>
<ul>
<li>Cartesian product:
<ul>
<li><span class="math inline">\(A \times B = \{ (p, q) : p \in A \land q \in B \}\)</span></li>
</ul></li>
<li>Def: A relation <span class="math inline">\(R\)</span> on a set <span class="math inline">\(\mathbb X\)</span> is a subset of <span class="math inline">\(\mathbb X \times \mathbb X\)</span> (<span class="math inline">\(R \subseteq \mathbb X \times \mathbb X\)</span>)</li>
<li>Graph of a function <span class="math inline">\(f()\)</span>: <span class="math inline">\(\{ (x, f(x) : x \in Dom(f) \}\)</span></li>
<li>Properties of:
<ul>
<li>Reflexivity: <span class="math inline">\((\forall x \in \mathbb X ) (x, x) \in R \equiv (\forall x \in \mathbb X) x R x\)</span></li>
<li>Symmetricity: <span class="math inline">\([ (\forall x, y \in \mathbb X) (x, y) \in R \implies (y, x) \in R) ] \equiv [ (\forall x, y \in \mathbb X) ( x R y \implies y R x) ]\)</span></li>
<li>Transitivity: <span class="math inline">\((\forall x, y, z \in \mathbb X) (x R y \land y R z \implies x R z)\)</span></li>
<li>Antisymmetricity: <span class="math inline">\((\forall x, y \in \mathbb X) (x R y \land y R x \implies x = y)\)</span></li>
</ul></li>
<li>Equivalence relations:
<ul>
<li>Def: <span class="math inline">\(R \subseteq \mathbb X \times \mathbb X\)</span> is said to be an equivalence relation <em>iff</em> <span class="math inline">\(R\)</span> is reflexive, symmetric and transitive.</li>
<li>Congruence modulo n: <span class="math inline">\(p R q \equiv n | p - q\)</span></li>
<li>Def R - and equivalence relation of <span class="math inline">\(\mathbb X\)</span>: The <em>equivalence class</em> of an element <span class="math inline">\(x \in \mathbb X\)</span> is the set <span class="math inline">\([x]_R = \{ y \in \mathbb X : x R y \}\)</span>
<ul>
<li>Every <span class="math inline">\(x \in \mathbb X\)</span> belongs to the equivalence class of some element <span class="math inline">\(a\)</span>.</li>
<li><span class="math inline">\((\forall x, y \in \mathbb X) ([x] \cap [y] \neq \emptyset \iff [x] = [y])\)</span></li>
</ul></li>
</ul></li>
<li>Partitions
<ul>
<li>A partition is a set containing subsets of some set <span class="math inline">\(\mathbb X\)</span> such that their collective symmetric difference equals <span class="math inline">\(\mathbb X\)</span>. A partition of is a set <span class="math inline">\(\{ A_i: i \in \mathbb I \land A_i \subseteq \mathbb X \}\)</span> such that:
<ul>
<li><span class="math inline">\((\forall x \in \mathbb X) (\exists j \in \mathbb I) (x \in A_j)\)</span></li>
<li><span class="math inline">\((\forall i, j \in \mathbb I) (i \neq j \implies A_i \cap A_j = \emptyset)\)</span></li>
</ul></li>
<li><span class="math inline">\(\{ A_i \}_{i \in \mathbb I}\)</span> is a partition <em>iff</em> there exists an equivalence relation <span class="math inline">\(R\)</span> on <span class="math inline">\(\mathbb X\)</span> such that:
<ul>
<li><span class="math inline">\((\forall i \in \mathbb I) (\exists x \in \mathbb X) A_i = [x]_R\)</span></li>
<li><span class="math inline">\((\forall x \in \mathbb X) (\exists j \in \mathbb I) [x] = A_j\)</span></li>
</ul></li>
<li>The quotient set: <span class="math inline">\(\mathbb X / R = \{ [a] : a \in \mathbb X \}\)</span></li>
</ul></li>
</ul>
<h2 id="posets">Posets<a href="#posets" class="hanchor" ariaLabel="Anchor">&#8983;</a> </h2>
<ul>
<li>Partial orders
<ul>
<li><span class="math inline">\(\mathbb X\)</span> is a set, <span class="math inline">\(R \subseteq \mathbb X \times \mathbb X\)</span></li>
<li>Def: <span class="math inline">\(R\)</span> is a partial order on <span class="math inline">\(\mathbb X\)</span> iff <span class="math inline">\(R\)</span> is:
<ul>
<li>Reflexive</li>
<li>Antisymmetric</li>
<li>Transitive</li>
</ul></li>
<li>Def: <span class="math inline">\(m \in \mathbb X\)</span> is said to be:
<ul>
<li>maximal element in <span class="math inline">\((\mathbb X, \preccurlyeq)\)</span> iff <span class="math inline">\((\forall a \in \mathbb X) m \preccurlyeq a \implies m = a\)</span></li>
<li>largest iff <span class="math inline">\((\forall a \in \mathbb X) (a \preccurlyeq m)\)</span></li>
<li>minimal iff <span class="math inline">\((\forall a \in \mathbb X) (a \preccurlyeq m \implies a = m)\)</span></li>
<li>smallest iff <span class="math inline">\((\forall a \in \mathbb X) (m \preccurlyeq a)\)</span></li>
</ul></li>
<li>Def: A partial order <span class="math inline">\(R\)</span> on <span class="math inline">\(\mathbb X\)</span> is said to be <em>“total”</em> iff <span class="math inline">\((\forall x, y \in \mathbb X) (x R y \lor y R x)\)</span></li>
<li>Def: A subset <span class="math inline">\(B\)</span> of <span class="math inline">\(\mathbb X\)</span> is called a chain <em>“chain”</em> iff <span class="math inline">\(B\)</span> is totally ordered by <span class="math inline">\(R\)</span>
<ul>
<li><span class="math inline">\(C(\mathbb X)\)</span> - the set of all chains in <span class="math inline">\((\mathbb X, R)\)</span></li>
<li>A chain <span class="math inline">\(D\)</span> in <span class="math inline">\((\mathbb X, R)\)</span> is called a maximal chain iff <span class="math inline">\(D\)</span> is a maximal element in <span class="math inline">\((C(\mathbb X), R)\)</span></li>
<li><span class="math inline">\(K \subseteq \mathbb X\)</span> is called an antichain in <span class="math inline">\((\mathbb X, R)\)</span> iff <span class="math inline">\((\forall p, q \in K) (p R q \lor q R p \implies p = q)\)</span></li>
<li>Def: <span class="math inline">\(R\)</span> is a partial order on <span class="math inline">\(\mathbb X\)</span>, <span class="math inline">\(R\)</span> is called a <em>well</em> order iff <span class="math inline">\(R\)</span> is a total order on <span class="math inline">\(X\)</span> and every nonempty subset <span class="math inline">\(A\)</span> of <span class="math inline">\(\mathbb X\)</span> has the smallest element</li>
</ul></li>
</ul></li>
</ul>
<h2 id="induction">Induction<a href="#induction" class="hanchor" ariaLabel="Anchor">&#8983;</a> </h2>
<ul>
<li>If <span class="math inline">\(\phi\)</span> is a propositional function defined on <span class="math inline">\(\mathbb N\)</span>, if:
<ul>
<li><span class="math inline">\(\phi(1)\)</span></li>
<li><span class="math inline">\((\forall n \geq 1) \phi(n) \implies \phi(n+1)\)</span></li>
<li><span class="math inline">\((\forall k \geq 1) \phi(k)\)</span></li>
</ul></li>
</ul>
<h2 id="functions">Functions<a href="#functions" class="hanchor" ariaLabel="Anchor">&#8983;</a> </h2>
<ul>
<li><span class="math inline">\(f: \mathbb X \to \mathbb Y\)</span></li>
<li>Def: <span class="math inline">\(f \subseteq \mathbb X \times \mathbb Y\)</span> is said to be a function if:
<ul>
<li><span class="math inline">\((\forall x \in \mathbb X)(\exists y \in \mathbb Y) (x, y) \in f(y = f(x))\)</span></li>
<li><span class="math inline">\((\forall a \in \mathbb X)(\forall p, q \in \mathbb Y)((a, p) \in f \land (a, q) \in f \implies p = q)\)</span></li>
</ul></li>
<li>Types of functions <span class="math inline">\(f: \mathbb X \to \mathbb Y\)</span>:
<ul>
<li><span class="math inline">\(f\)</span> is said to be an injection ( 1 to 1 function) iff <span class="math inline">\((\forall x_1, x_2 \in \mathbb X) x_1 \neq x_2 \implies f(x_1) \neq f(x_2)\)</span></li>
<li><span class="math inline">\(f\)</span> is said to be a surjection (onto function) iff <span class="math inline">\((\forall y \in \mathbb Y)(\exists x \in \mathbb X) f(x) = y\)</span></li>
<li>If <span class="math inline">\(f^{-1}\)</span> is a function from <span class="math inline">\(\mathbb Y \to \mathbb X\)</span> then <span class="math inline">\(f^{-1}\)</span> is called the inverse function for <span class="math inline">\(f\)</span>
<ul>
<li>Fact: <span class="math inline">\(f^{-1}\)</span> is a function iff <span class="math inline">\(f\)</span> is a <em>bijection</em> (1 to 1 and onto)</li>
</ul></li>
</ul></li>
<li>For some set <span class="math inline">\(\mathbb A\)</span> the image of <span class="math inline">\(\mathbb A\)</span> by <span class="math inline">\(f\)</span> is <span class="math inline">\(f(\mathbb A) = \{ f(x) : x \in \mathbb A \}\)</span>. We can also define the inverse of an image even when the function itself isn’t invertible: <span class="math inline">\(f^{-1}(\mathbb A)\)</span></li>
</ul>
<h2 id="combinatorics">Combinatorics<a href="#combinatorics" class="hanchor" ariaLabel="Anchor">&#8983;</a> </h2>
<ul>
<li><span class="math inline">\(|\mathbb A|\)</span> size (number of elements) of <span class="math inline">\(\mathbb A\)</span></li>
<li>Rule of addition:
<ul>
<li>If <span class="math inline">\(\mathbb A, \mathbb B \subseteq \mathbb X\)</span> and <span class="math inline">\(|\mathbb A|, |\mathbb B| \in \mathbb N\)</span> and <span class="math inline">\(\mathbb A \cap \mathbb B = \emptyset\)</span> then <span class="math inline">\(|\mathbb A \cup \mathbb B| = |\mathbb A| + |\mathbb B|\)</span></li>
<li>Can be generalized as: <span class="math display">\[
(\forall n ) \mathbb{A}_1, \mathbb{A}_2, ..., \mathbb{A}_n \in \mathbb{X} \land \\
|\mathbb{A}_1|, |\mathbb{A}_2|, ..., |\mathbb{A}_n| \in \mathbb{N} \implies \\
(\forall i, j \in \{1, 2, ..., n \})(i \neq j \implies \mathbb{A}_i \cap \mathbb{A}_j = \emptyset)
\]</span></li>
</ul></li>
<li>Rule of multiplication:
<ul>
<li><span class="math inline">\(\mathbb{A}, \mathbb{B} \subseteq \mathbb{X}, |\mathbb{A} \times \mathbb{B}| = |\mathbb{A}| \cdot |\mathbb{B}|\)</span></li>
<li>Can be generalized as: <span class="math display">\[
(\forall n ) \mathbb{A}_1, \mathbb{A}_2, ..., \mathbb{A}_n \in \mathbb{X} \land |\mathbb{A}_i| \in \mathbb{N} \implies \\
|\mathbb{A}_1 \times \mathbb{A}_2 \times ... \times \mathbb{A}_n| = |\mathbb{A}_1| \cdot |\mathbb{A}_2| \cdot ... \cdot |\mathbb{A_n}|
\]</span></li>
</ul></li>
</ul>
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Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets. Propositional calculus Comes from the linguistic concept that things can be either true or false.
We should avoid variables when forming statements, as they may change the logical value.
(2=7) statement (x=5) not a statement In logic we do not use the equals sign, we use the equivalence sign (\equiv).
Logical values (booleans) are denoted by either 0 or 1 (or t, f, etc.
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<title>Introduction to Discrete Mathematics</title>
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<pubDate>Wed, 20 Nov 2019 00:00:00 +0000</pubDate>
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<description>Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets. Propositional calculus Comes from the linguistic concept that things can be either true or false.
We should avoid variables when forming statements, as they may change the logical value.
\(2=7\) statement \(x=5\) not a statement In logic we do not use the equals sign, we use the equivalence sign \(\equiv\).
Logical values (booleans) are denoted by either 0 or 1 (or t, f, etc.</description>
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