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Signed-off-by: Abdulkadir Furkan Şanlı <abdulkadirfsanli@protonmail.com>
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Abdulkadir Furkan Şanlı 2019-11-07 10:20:17 +01:00
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<li><p>The quotient set: <span class="math">\(\mathbb X / R = \{ [a] : a \in \mathbb X \}\)</span></p></li> <li><p>The quotient set: <span class="math">\(\mathbb X / R = \{ [a] : a \in \mathbb X \}\)</span></p></li>
</ul></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> </ul>
</div> </div>