From 91a8f3a61890c458e73c727394c9022005f5bd2f Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Abdulkadir=20Furkan=20=C5=9Eanl=C4=B1?=
 <abdulkadirfsanli@protonmail.com>
Date: Thu, 7 Nov 2019 10:20:17 +0100
Subject: [PATCH] Regen site
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

Signed-off-by: Abdulkadir Furkan Şanlı <abdulkadirfsanli@protonmail.com>
---
 public/posts/eidma/index.html | 50 +++++++++++++++++++++++++++++++++++
 1 file changed, 50 insertions(+)

diff --git a/public/posts/eidma/index.html b/public/posts/eidma/index.html
index b7028dd..18074c0 100644
--- a/public/posts/eidma/index.html
+++ b/public/posts/eidma/index.html
@@ -384,6 +384,56 @@
 
 <li><p>The quotient set: <span  class="math">\(\mathbb X / R = \{ [a] : a \in \mathbb X \}\)</span></p></li>
 </ul></li>
+</ul>
+
+<h2 id="posets">Posets</h2>
+
+<ul>
+<li><p>Partial orders</p>
+
+<ul>
+<li><span  class="math">\(\mathbb X\)</span> is a set, <span  class="math">\(R \subseteq \mathbb X \times \mathbb X\)</span></li>
+
+<li><p>Def: <span  class="math">\(R\)</span> is a partial order on <span  class="math">\(\mathbb X\)</span> iff <span  class="math">\(R\)</span> is:</p>
+
+<ul>
+<li>Reflexive</li>
+<li>Antisymmetric</li>
+<li>Transitive</li>
+</ul></li>
+
+<li><p>Def: <span  class="math">\(m \in \mathbb X\)</span> is said to be:</p>
+
+<ul>
+<li>maximal element in <span  class="math">\((\mathbb X, \preccurlyeq)\)</span> iff <span  class="math">\((\forall a \in \mathbb X) m \preccurlyeq a \implies m = a\)</span></li>
+<li>largest iff <span  class="math">\((\forall a \in \mathbb X) (a \preccurlyeq m)\)</span></li>
+<li>minimal iff <span  class="math">\((\forall a \in \mathbb X) (a \preccurlyeq m \implies a = m)\)</span></li>
+<li>smallest iff <span  class="math">\((\forall a \in \mathbb X) (m \preccurlyeq a)\)</span></li>
+</ul></li>
+
+<li><p>Def: A partial order <span  class="math">\(R\)</span> on <span  class="math">\(\mathbb X\)</span> is said to be <em>&quot;total&quot;</em> iff <span  class="math">\((\forall x, y \in \mathbb X) (x R y \lor y R x)\)</span></p></li>
+
+<li><p>Def: A subset <span  class="math">\(B\)</span> of <span  class="math">\(\mathbb X\)</span> is called a chain <em>&quot;chain&quot;</em> iff <span  class="math">\(B\)</span> is totally ordered by <span  class="math">\(R\)</span></p>
+
+<ul>
+<li><span  class="math">\(C(\mathbb X)\)</span> - the set of all chains in <span  class="math">\((\mathbb X, R)\)</span></li>
+<li>A chain <span  class="math">\(D\)</span> in <span  class="math">\((\mathbb X, R)\)</span> is called a maximal chain iff <span  class="math">\(D\)</span> is a maximal element in <span  class="math">\((C(\mathbb X), R)\)</span></li>
+<li><span  class="math">\(K \subseteq \mathbb X\)</span> is called an antichain in <span  class="math">\((\mathbb X, R)\)</span> iff <span  class="math">\((\forall p, q \in K) (p R q \lor q R p \implies p = q)\)</span></li>
+<li>Def: <span  class="math">\(R\)</span> is a partial order on <span  class="math">\(\mathbb X\)</span>, <span  class="math">\(R\)</span> is called a <em>well</em> order iff <span  class="math">\(R\)</span> is a total order on <span  class="math">\(X\)</span> and every nonempty subset <span  class="math">\(A\)</span> of <span  class="math">\(\mathbb X\)</span> has the smallest element</li>
+</ul></li>
+</ul></li>
+</ul>
+
+<h2 id="induction">Induction</h2>
+
+<ul>
+<li>If <span  class="math">\(\phi\)</span> is a propositional function defined on <span  class="math">\(\mathbb N\)</span>, if:
+
+<ul>
+<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">\((\forall k \geq 1) \phi(k)\)</span></li>
+</ul></li>
 </ul>
 
   </div>