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Update eidma

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Signed-off-by: Abdulkadir Furkan Şanlı <abdulkadirfsanli@protonmail.com>
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Abdulkadir Furkan Şanlı
2019-11-20 11:07:13 +01:00
parent 6744f6ffc1
commit e6290c0bd2
10 changed files with 94 additions and 61 deletions
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@ -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 \(\equiv\).
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:creator" content="Abdulkadir" />
<meta name="twitter:creator" content="" />
<meta name="twitter:image" content="">
@ -51,7 +51,7 @@
<meta property="og:image:width" content="2048">
<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">
<span class="post-date">
2019-11-04
2019-11-20
</span>
<span class="post-author">::
Abdulkadir
</span>
</div>
@ -431,9 +427,36 @@
<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">\((\forall n \geq 1) \phi(n) \implies \phi(n+1)\)</span></li>
<li><span class="math">\((\forall k \geq 1) \phi(k)\)</span></li>
</ul></li>
</ul>
<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>
</div>