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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
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title = "Introduction to Discrete Mathematics"
date = "2019-11-04"
author = "Abdulkadir"
showFullContent = false
date = "2019-11-20"
tags = ["university-notes"]
markup = "mmark"
+++
- 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
## Induction
- If $$\phi$$ is a propositional function defined on $$\mathbb N$$, if:
- $$\phi(1)$$
- $$(\forall n \geq 1) (\phi(n) \implies \phi(n+1)$$
- $$(\forall n \geq 1) \phi(n) \implies \phi(n+1)$$
- $$(\forall k \geq 1) \phi(k)$$
## 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: $$f^{-1}(\mathbb A)$$

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

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<generator>Hugo -- gohugo.io</generator>
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<copyright>© Abdulkadir Furkan Şanlı 2019</copyright>
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<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>
<title>about</title>
<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>
</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>
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\(2=7\) 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" />
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<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>

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

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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.

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

4
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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.