afk/personal-site
1
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

Update eidma

Browse Source 
Signed-off-by: Abdulkadir Furkan Şanlı <abdulkadirfsanli@protonmail.com>
This commit is contained in:
Abdulkadir Furkan Şanlı 2019-11-20 11:07:13 +01:00
parent 6744f6ffc1
commit e6290c0bd2
No known key found for this signature in database
GPG Key ID: 7823BD18E6F95D73
10 changed files with 94 additions and 61 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

23
content/posts/eidma.mmark
View File

@ -1,10 +1,7 @@
+++ +++
title = "Introduction to Discrete Mathematics" title = "Introduction to Discrete Mathematics"
date = "2019-11-04" date = "2019-11-20"
author = "Abdulkadir"
showFullContent = false
tags = ["university-notes"] tags = ["university-notes"]
markup = "mmark"
+++ +++
- Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets. - Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets.
@ -161,7 +158,23 @@ markup = "mmark"
- Def: $$R$$ is a partial order on $$\mathbb X$$, $$R$$ is called a *well* order iff $$R$$ is a total order on $$X$$ and every nonempty subset $$A$$ of $$\mathbb X$$ has the smallest element - Def: $$R$$ is a partial order on $$\mathbb X$$, $$R$$ is called a *well* order iff $$R$$ is a total order on $$X$$ and every nonempty subset $$A$$ of $$\mathbb X$$ has the smallest element
## Induction ## Induction
- If $$\phi$$ is a propositional function defined on $$\mathbb N$$, if: - If $$\phi$$ is a propositional function defined on $$\mathbb N$$, if:
- $$\phi(1)$$ - $$\phi(1)$$
- $$(\forall n \geq 1) (\phi(n) \implies \phi(n+1)$$ - $$(\forall n \geq 1) \phi(n) \implies \phi(n+1)$$
- $$(\forall k \geq 1) \phi(k)$$ - $$(\forall k \geq 1) \phi(k)$$
## Functions
- $$f: \mathbb X \to \mathbb Y$$
- Def: $$f \subseteq \mathbb X \times \mathbb Y$$ is said to be a function if:
- $$(\forall x \in \mathbb X)(\exists y \in \mathbb Y) (x, y) \in f(y = f(x))$$
- $$(\forall a \in \mathbb X)(\forall p, q \in \mathbb Y)((a, p) \in f \land (a, q) \in f \implies p = q)$$
- Types of functions $$f: \mathbb X \to \mathbb Y$$:
- $$f$$ is said to be an injection ( 1 to 1 function) iff $$(\forall x_1, x_2 \in \mathbb X) x_1 \neq x_2 \implies f(x_1) \neq f(x_2)$$
- $$f$$ is said to be a surjection (onto function) iff $$(\forall y \in \mathbb Y)(\exists x \in \mathbb X) f(x) = y$$
- If $$f^{-1}$$ is a function from $$\mathbb Y \to \mathbb X$$ then $$f^{-1}$$ is called the inverse function for $$f$$
- Fact: $$f^{-1}$$ is a function iff $$f$$ is a *bijection* (1 to 1 and onto)
- For some set $$\mathbb A$$ the image of $$\mathbb A$$ by $$f$$ is $$f(\mathbb A) = \{ f(x) : x \in \mathbb A \}$$. We can also define the inverse of an image even when the function itself isn't invertible: $$f^{-1}(\mathbb A)$$

5
public/index.html
View File

@ -141,10 +141,9 @@
<a href="https://abdulocra.cy/posts/eidma/">Introduction to Discrete Mathematics</a></h1> <a href="https://abdulocra.cy/posts/eidma/">Introduction to Discrete Mathematics</a></h1>
<div class="post-meta"> <div class="post-meta">
<span class="post-date"> <span class="post-date">
2019-11-04 2019-11-20
</span> </span>
<span class="post-author">::
Abdulkadir</span>
</div> </div>

24
public/index.xml
View File

@ -7,11 +7,22 @@
<generator>Hugo -- gohugo.io</generator> <generator>Hugo -- gohugo.io</generator>
<language>en-us</language> <language>en-us</language>
<copyright>© Abdulkadir Furkan Şanlı 2019</copyright> <copyright>© Abdulkadir Furkan Şanlı 2019</copyright>
<lastBuildDate>Mon, 04 Nov 2019 11:14:55 +0100</lastBuildDate> <lastBuildDate>Wed, 20 Nov 2019 00:00:00 +0000</lastBuildDate>
<atom:link href="https://abdulocra.cy/index.xml" rel="self" type="application/rss+xml" /> <atom:link href="https://abdulocra.cy/index.xml" rel="self" type="application/rss+xml" />
<item>
<title>Introduction to Discrete Mathematics</title>
<link>https://abdulocra.cy/posts/eidma/</link>
<pubDate>Wed, 20 Nov 2019 00:00:00 +0000</pubDate>
<guid>https://abdulocra.cy/posts/eidma/</guid>
<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>
</item>
<item> <item>
<title>about</title> <title>about</title>
<link>https://abdulocra.cy/about/</link> <link>https://abdulocra.cy/about/</link>
@ -21,16 +32,5 @@
<description> name: Abdulkadir Furkan Şanlı handle: abdulocracy contact: email: my handle at disroot dot org irc (freenode): abdulocracy </description> <description> name: Abdulkadir Furkan Şanlı handle: abdulocracy contact: email: my handle at disroot dot org irc (freenode): abdulocracy </description>
</item> </item>
<item>
<title>Introduction to Discrete Mathematics</title>
<link>https://abdulocra.cy/posts/eidma/</link>
<pubDate>Mon, 04 Nov 2019 00:00:00 +0000</pubDate>
<guid>https://abdulocra.cy/posts/eidma/</guid>
<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>
</item>
</channel> </channel>
</rss> </rss>

39
public/posts/eidma/index.html
View File

@ -35,7 +35,7 @@
\(2=7\) statement \(x=5\) not a statement In logic we do not use the equals sign, we use the equivalence sign \(\equiv\). \(2=7\) 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." /> Logical values (booleans) are denoted by either 0 or 1 (or t, f, etc." />
<meta name="twitter:site" content="https://abdulocra.cy/" /> <meta name="twitter:site" content="https://abdulocra.cy/" />
<meta name="twitter:creator" content="Abdulkadir" /> <meta name="twitter:creator" content="" />
<meta name="twitter:image" content=""> <meta name="twitter:image" content="">
@ -51,7 +51,7 @@
<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="2019-11-04 00:00:00 &#43;0000 UTC" /> <meta property="article:published_time" content="2019-11-20 00:00:00 &#43;0000 UTC" />
@ -134,14 +134,10 @@
<div class="post-meta"> <div class="post-meta">
<span class="post-date"> <span class="post-date">
2019-11-04 2019-11-20
</span> </span>
<span class="post-author">::
Abdulkadir
</span>
</div> </div>
@ -431,9 +427,36 @@
<ul> <ul>
<li><span class="math">\(\phi(1)\)</span></li> <li><span class="math">\(\phi(1)\)</span></li>
<li><span class="math">\((\forall n \geq 1) (\phi(n) \implies \phi(n+1)\)</span></li> <li><span class="math">\((\forall n \geq 1) \phi(n) \implies \phi(n+1)\)</span></li>
<li><span class="math">\((\forall k \geq 1) \phi(k)\)</span></li> <li><span class="math">\((\forall k \geq 1) \phi(k)\)</span></li>
</ul></li> </ul></li>
</ul>
<h2 id="functions">Functions</h2>
<ul>
<li><span class="math">\(f: \mathbb X \to \mathbb Y\)</span></li>
<li><p>Def: <span class="math">\(f \subseteq \mathbb X \times \mathbb Y\)</span> is said to be a function if:</p>
<ul>
<li><span class="math">\((\forall x \in \mathbb X)(\exists y \in \mathbb Y) (x, y) \in f(y = f(x))\)</span></li>
<li><span class="math">\((\forall a \in \mathbb X)(\forall p, q \in \mathbb Y)((a, p) \in f \land (a, q) \in f \implies p = q)\)</span></li>
</ul></li>
<li><p>Types of functions <span class="math">\(f: \mathbb X \to \mathbb Y\)</span>:</p>
<ul>
<li><span class="math">\(f\)</span> is said to be an injection ( 1 to 1 function) iff <span class="math">\((\forall x_1, x_2 \in \mathbb X) x_1 \neq x_2 \implies f(x_1) \neq f(x_2)\)</span></li>
<li><span class="math">\(f\)</span> is said to be a surjection (onto function) iff <span class="math">\((\forall y \in \mathbb Y)(\exists x \in \mathbb X) f(x) = y\)</span></li>
<li>If <span class="math">\(f^{-1}\)</span> is a function from <span class="math">\(\mathbb Y \to \mathbb X\)</span> then <span class="math">\(f^{-1}\)</span> is called the inverse function for <span class="math">\(f\)</span>
<ul>
<li>Fact: <span class="math">\(f^{-1}\)</span> is a function iff <span class="math">\(f\)</span> is a <em>bijection</em> (1 to 1 and onto)</li>
</ul></li>
</ul></li>
<li><p>For some set <span class="math">\(\mathbb A\)</span> the image of <span class="math">\(\mathbb A\)</span> by <span class="math">\(f\)</span> is <span class="math">\(f(\mathbb A) = \{ f(x) : x \in \mathbb A \}\)</span>. We can also define the inverse of an image even when the function itself isn't invertible: <span class="math">\(f^{-1}(\mathbb A)\)</span></p></li>
</ul> </ul>
</div> </div>

5
public/posts/index.html
View File

@ -138,10 +138,9 @@
<a href="https://abdulocra.cy/posts/eidma/">Introduction to Discrete Mathematics</a></h1> <a href="https://abdulocra.cy/posts/eidma/">Introduction to Discrete Mathematics</a></h1>
<div class="post-meta"> <div class="post-meta">
<span class="post-date"> <span class="post-date">
2019-11-04 2019-11-20
</span> </span>
<span class="post-author">::
Abdulkadir</span>
</div> </div>

4
public/posts/index.xml
View File

@ -7,7 +7,7 @@
<generator>Hugo -- gohugo.io</generator> <generator>Hugo -- gohugo.io</generator>
<language>en-us</language> <language>en-us</language>
<copyright>© Abdulkadir Furkan Şanlı 2019</copyright> <copyright>© Abdulkadir Furkan Şanlı 2019</copyright>
<lastBuildDate>Mon, 04 Nov 2019 00:00:00 +0000</lastBuildDate> <lastBuildDate>Wed, 20 Nov 2019 00:00:00 +0000</lastBuildDate>
<atom:link href="https://abdulocra.cy/posts/index.xml" rel="self" type="application/rss+xml" /> <atom:link href="https://abdulocra.cy/posts/index.xml" rel="self" type="application/rss+xml" />
@ -15,7 +15,7 @@
<item> <item>
<title>Introduction to Discrete Mathematics</title> <title>Introduction to Discrete Mathematics</title>
<link>https://abdulocra.cy/posts/eidma/</link> <link>https://abdulocra.cy/posts/eidma/</link>
<pubDate>Mon, 04 Nov 2019 00:00:00 +0000</pubDate> <pubDate>Wed, 20 Nov 2019 00:00:00 +0000</pubDate>
<guid>https://abdulocra.cy/posts/eidma/</guid> <guid>https://abdulocra.cy/posts/eidma/</guid>
<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. <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.

42
public/sitemap.xml
View File

@ -4,7 +4,27 @@
<url> <url>
<loc>https://abdulocra.cy/</loc> <loc>https://abdulocra.cy/</loc>
<lastmod>2019-11-04T11:14:55+01:00</lastmod> <lastmod>2019-11-20T00:00:00+00:00</lastmod>
</url>
<url>
<loc>https://abdulocra.cy/posts/eidma/</loc>
<lastmod>2019-11-20T00:00:00+00:00</lastmod>
</url>
<url>
<loc>https://abdulocra.cy/posts/</loc>
<lastmod>2019-11-20T00:00:00+00:00</lastmod>
</url>
<url>
<loc>https://abdulocra.cy/tags/</loc>
<lastmod>2019-11-20T00:00:00+00:00</lastmod>
</url>
<url>
<loc>https://abdulocra.cy/tags/university-notes/</loc>
<lastmod>2019-11-20T00:00:00+00:00</lastmod>
</url> </url>
<url> <url>
@ -12,26 +32,6 @@
<lastmod>2019-11-04T11:14:55+01:00</lastmod> <lastmod>2019-11-04T11:14:55+01:00</lastmod>
</url> </url>
<url>
<loc>https://abdulocra.cy/posts/eidma/</loc>
<lastmod>2019-11-04T00:00:00+00:00</lastmod>
</url>
<url>
<loc>https://abdulocra.cy/posts/</loc>
<lastmod>2019-11-04T00:00:00+00:00</lastmod>
</url>
<url>
<loc>https://abdulocra.cy/tags/</loc>
<lastmod>2019-11-04T00:00:00+00:00</lastmod>
</url>
<url>
<loc>https://abdulocra.cy/tags/university-notes/</loc>
<lastmod>2019-11-04T00:00:00+00:00</lastmod>
</url>
<url> <url>
<loc>https://abdulocra.cy/categories/</loc> <loc>https://abdulocra.cy/categories/</loc>
</url> </url>

4
public/tags/index.xml
View File

@ -7,7 +7,7 @@
<generator>Hugo -- gohugo.io</generator> <generator>Hugo -- gohugo.io</generator>
<language>en-us</language> <language>en-us</language>
<copyright>© Abdulkadir Furkan Şanlı 2019</copyright> <copyright>© Abdulkadir Furkan Şanlı 2019</copyright>
<lastBuildDate>Mon, 04 Nov 2019 00:00:00 +0000</lastBuildDate> <lastBuildDate>Wed, 20 Nov 2019 00:00:00 +0000</lastBuildDate>
<atom:link href="https://abdulocra.cy/tags/index.xml" rel="self" type="application/rss+xml" /> <atom:link href="https://abdulocra.cy/tags/index.xml" rel="self" type="application/rss+xml" />
@ -15,7 +15,7 @@
<item> <item>
<title>university-notes</title> <title>university-notes</title>
<link>https://abdulocra.cy/tags/university-notes/</link> <link>https://abdulocra.cy/tags/university-notes/</link>
<pubDate>Mon, 04 Nov 2019 00:00:00 +0000</pubDate> <pubDate>Wed, 20 Nov 2019 00:00:00 +0000</pubDate>
<guid>https://abdulocra.cy/tags/university-notes/</guid> <guid>https://abdulocra.cy/tags/university-notes/</guid>
<description></description> <description></description>

5
public/tags/university-notes/index.html
View File

@ -138,10 +138,9 @@
<a href="https://abdulocra.cy/posts/eidma/">Introduction to Discrete Mathematics</a></h1> <a href="https://abdulocra.cy/posts/eidma/">Introduction to Discrete Mathematics</a></h1>
<div class="post-meta"> <div class="post-meta">
<span class="post-date"> <span class="post-date">
2019-11-04 2019-11-20
</span> </span>
<span class="post-author">::
Abdulkadir</span>
</div> </div>

4
public/tags/university-notes/index.xml
View File

@ -7,7 +7,7 @@
<generator>Hugo -- gohugo.io</generator> <generator>Hugo -- gohugo.io</generator>
<language>en-us</language> <language>en-us</language>
<copyright>© Abdulkadir Furkan Şanlı 2019</copyright> <copyright>© Abdulkadir Furkan Şanlı 2019</copyright>
<lastBuildDate>Mon, 04 Nov 2019 00:00:00 +0000</lastBuildDate> <lastBuildDate>Wed, 20 Nov 2019 00:00:00 +0000</lastBuildDate>
<atom:link href="https://abdulocra.cy/tags/university-notes/index.xml" rel="self" type="application/rss+xml" /> <atom:link href="https://abdulocra.cy/tags/university-notes/index.xml" rel="self" type="application/rss+xml" />
@ -15,7 +15,7 @@
<item> <item>
<title>Introduction to Discrete Mathematics</title> <title>Introduction to Discrete Mathematics</title>
<link>https://abdulocra.cy/posts/eidma/</link> <link>https://abdulocra.cy/posts/eidma/</link>
<pubDate>Mon, 04 Nov 2019 00:00:00 +0000</pubDate> <pubDate>Wed, 20 Nov 2019 00:00:00 +0000</pubDate>
<guid>https://abdulocra.cy/posts/eidma/</guid> <guid>https://abdulocra.cy/posts/eidma/</guid>
<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. <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.