<!DOCTYPE html>
<html lang="en">
<head>
<title>Introduction to Discrete Mathematics :: abdulocracy's personal site</title>
<meta http-equiv="content-type" content="text/html; charset=utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0, maximum-scale=1">
<meta name="description" content="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."/>
<meta name="keywords" content=""/>
<meta name="robots" content="noodp"/>
<link rel="canonical" href="https://022385.xyz/posts/eidma/" />
<link rel="stylesheet" href="https://022385.xyz/assets/style.css">
<link rel="stylesheet" href="https://022385.xyz/assets/pink.css">
<link rel="apple-touch-icon-precomposed" sizes="144x144" href="https://022385.xyz/img/apple-touch-icon-144-precomposed.png">
<link rel="shortcut icon" href="https://022385.xyz/favicon.png">
<meta name="twitter:card" content="summary" />
<meta name="twitter:title" content="Introduction to Discrete Mathematics :: abdulocracy's personal site — " />
<meta name="twitter:description" content="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." />
<meta name="twitter:site" content="https://022385.xyz/" />
<meta name="twitter:creator" content="Abdulkadir" />
<meta name="twitter:image" content="">
<meta property="og:locale" content="en" />
<meta property="og:type" content="article" />
<meta property="og:title" content="Introduction to Discrete Mathematics :: abdulocracy's personal site — ">
<meta property="og:description" content="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." />
<meta property="og:url" content="https://022385.xyz/posts/eidma/" />
<meta property="og:site_name" content="Introduction to Discrete Mathematics" />
<meta property="og:image" content="">
<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 +0000 UTC" />
</head>
<body class="">
<div class="container center">
<header class="header">
<div class="header__inner">
<div class="header__logo">
<a href="/">
<div class="logo">
abdulocracy
</div>
</a>
</div>
<div class="menu-trigger">menu</div>
</div>
<nav class="menu">
<ul class="menu__inner menu__inner--desktop">
<ul class="menu__sub-inner">
<li class="menu__sub-inner-more-trigger">≡ ▾</li>
<ul class="menu__sub-inner-more hidden">
<li><a href="/about">about</a></li>
<li><a href="/tags/university-notes">university notes</a></li>
</ul>
</ul>
</ul>
<ul class="menu__inner menu__inner--mobile">
<li><a href="/about">about</a></li>
<li><a href="/tags/university-notes">university notes</a></li>
</ul>
</nav>
</header>
<div class="content">
<div class="post">
<h1 class="post-title">
<a href="https://022385.xyz/posts/eidma/">Introduction to Discrete Mathematics</a></h1>
<div class="post-meta">
<span class="post-date">
2019-11-04
</span>
<span class="post-author">::
Abdulkadir
</span>
</div>
<span class="post-tags">
#<a href="https://022385.xyz/tags/university-notes/">university-notes</a>
</span>
<div class="post-content">
<ul>
<li>Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets.</li>
</ul>
<h2 id="propositional-calculus">Propositional calculus</h2>
<ul>
<li>Comes from the linguistic concept that things can be either true or false.</li>
<li><p>We should avoid variables when forming statements, as they may change the logical value.</p>
<ul>
<li><span class="math">\(2=7\)</span> statement</li>
<li><span class="math">\(x=5\)</span> not a statement</li>
</ul></li>
<li><p>In logic we do not use the equals sign, we use the equivalence sign <span class="math">\(\equiv\)</span>.</p></li>
<li><p>Logical values (booleans) are denoted by either 0 or 1 (or t, f, etc.).</p></li>
<li><p>When doing logic, we use propositional variables (e.g. p, q, r).</p>
<ul>
<li>Can be either <strong>true</strong> or <strong>false</strong>.</li>
</ul></li>
<li><p>The operations done on propositional variables are called propositional connectives.</p>
<ul>
<li>Conjunction: <span class="math">\(p \land q\)</span> is only true if both p and q are true <span class="math">\((0001)\)</span></li>
<li>Disjunction: <span class="math">\(p \lor q\)</span> is only false if both p and q are false <span class="math">\((0111)\)</span></li>
<li>Implication (material conditional): <span class="math">\(p \implies q\)</span> is false only if p is true and q is false (truth table <span class="math">\((1011)\)</span>)
<ul>
<li><span class="math">\(\equiv \neg p \lor q\)</span></li>
</ul></li>
</ul></li>
<li><p>Not necessarily connectives but unary operations:</p>
<ul>
<li>Negation: Denoted by ~, <span class="math">\(\neg\)</span> or NOT, negates the one input <span class="math">\((10)\)</span>.</li>
</ul></li>
<li><p>A (propositional) formula is a "properly constructed" logical expression.</p>
<ul>
<li>e.g. <span class="math">\(\neg[(p \lor q)] \land r\)</span></li>
<li><span class="math">\((p \land)\)</span> is not a formula, as <span class="math">\(\land\)</span> requires 2 variables.</li>
<li>Logical equivalence: <span class="math">\(\phi(p, q, k) \equiv \psi(p, q, k)\)</span>, logical value of <span class="math">\(\phi\)</span> is equal to logical value of <span class="math">\(\psi\)</span>.</li>
<li>Commutativity: <span class="math">\(p \land q \equiv q \land p\)</span></li>
<li>Associativity: <span class="math">\((p \land q) \land r \equiv p \land (q \land r)\)</span></li>
<li>Distributivity: <span class="math">\(p \land (q \lor r) \equiv (p \land q) \lor (p \land r)\)</span></li>
<li>Conjunctive normal form: every formula can be written as a conjunction of one or more disjunctions.
<ul>
<li><span class="math">\(\neg(B \lor C)\)</span> can be written as <span class="math">\(\neg B \land \neg C\)</span></li>
</ul></li>
</ul></li>
<li><p>Double negation law: <span class="math">\(\neg(\neg p) \equiv p\)</span></p></li>
<li><p>De Morgan's laws: <span class="math">\(\neg(p \land q) \equiv \neg p \lor \neg q\)</span> and <span class="math">\(\neg(p \lor q) \equiv \neg p \land \neg q\)</span>.</p></li>
<li><p>If and only if (<em>iff</em>): <span class="math">\(p \iff p \equiv (p \implies q) \land (q \implies p)\)</span></p></li>
<li><p>Contraposition law:</p>
<ul>
<li><span class="math">\((p \implies q) \equiv (\neg q \implies \neg p)\)</span> prove by contraposition
<ul>
<li><span class="math">\((p \implies q) \equiv (\neg p \lor q)\)</span></li>
<li><span class="math">\((\neg q \implies \neg p) \equiv (\neg (\neg q) \lor (\neg p) \equiv (q \lor \neg p) \equiv (\neg p \lor q)\)</span></li>
</ul></li>
</ul></li>
<li><p>Contradiction law:</p>
<ul>
<li><span class="math">\(p \lor \neg p \equiv 1\)</span> and <span class="math">\(p \land \neg p \equiv 0\)</span></li>
</ul></li>
<li><p>Tautology: <span class="math">\(\phi (p, q, ... r)\)</span> is a tautology <em>iff</em> <span class="math">\(\phi \equiv 1\)</span></p></li>
</ul>
<h2 id="sets">Sets</h2>
<ul>
<li><p>We will consider subsets of universal set <span class="math">\(\mathbb X\)</span></p>
<ul>
<li><span class="math">\(2^\mathbb X = \{ A : A \subseteq \mathbb X\}\)</span></li>
<li><span class="math">\(2^\mathbb X = P(\mathbb X)\)</span></li>
<li>All 2 object subsets of <span class="math">\(\mathbb X\)</span>: <span class="math">\(P_2(\mathbb X)\)</span></li>
</ul></li>
<li><p><span class="math">\(A \subset B \equiv\)</span> every element of A is an element of B <span class="math">\(\equiv \{x \in \mathbb X : x \in A \implies x \in B\}\)</span></p></li>
<li><p>Operations on sets:</p>
<ul>
<li>Union - <span class="math">\(\cup\)</span> - <span class="math">\(A \cup B = \{ x \in \mathbb X : x \in A \lor x \in B \}\)</span></li>
<li>Intersection - <span class="math">\(\cap\)</span> - <span class="math">\(A \cap B = \{ x \in \mathbb X : x \in A \land x \in B \}\)</span></li>
<li>Complement - <span class="math">\(A'\)</span> - <span class="math">\(A' = \{ x \in \mathbb X : \neg (x \in A) \}\)</span>
<ul>
<li>If <span class="math">\(x = \{ 1 \}\)</span> then <span class="math">\(x' = \emptyset\)</span></li>
</ul></li>
</ul></li>
<li><p>Equality of sets: <span class="math">\(A = B\)</span> iff <span class="math">\(x \in \mathbb X : (x \in A \iff x \in B)\)</span></p></li>
<li><p>Difference of sets:</p>
<ul>
<li><span class="math">\(A \setminus B = \{ x \in \mathbb X : x \in A \land x \notin B \} = A \cap B'\)</span></li>
<li>Symmetric difference: <span class="math">\(A \div B = (A \setminus B) \cup (B \setminus A)\)</span></li>
</ul></li>
<li><p>Laws of set algebra:</p>
<ul>
<li><span class="math">\(A \cup B = B \cup A , A \cap B = B \cap A\)</span></li>
<li><span class="math">\((A \cup B) \cup C = A \cup (B \cup C), (A \cap B) \cap C = A \cap (B \cap C)\)</span></li>
<li><span class="math">\((A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\)</span> vice versa</li>
<li><span class="math">\(A \cap \emptyset, A \cap \mathbb X = A, A \cup \emptyset = A, A \cup \mathbb X = \mathbb X\)</span></li>
<li><span class="math">\((A \cup B)' = A' \cap B'\)</span> vice versa</li>
<li><span class="math">\(A \cup A' = \mathbb X, A \cap A' = \emptyset\)</span></li>
</ul></li>
<li><p>Note: <span class="math">\(\{ \emptyset \} \neq \emptyset\)</span>, one is a set with one element, one is the empty set, no elements (<span class="math">\(\{ \}\)</span>)</p></li>
<li><p>Quip: <span class="math">\(\{ x \in \mathbb R : x^2 = -1\} = \emptyset\)</span></p></li>
</ul>
<h2 id="quantifiers">Quantifiers</h2>
<ul>
<li><span class="math">\(\phi\)</span> - prepositional function: yields only true or false value</li>
<li><span class="math">\(\forall\)</span> means "for all" and <span class="math">\(\exists\)</span> means "there exists"</li>
<li><p><span class="math">\(\forall\)</span>:</p>
<ul>
<li>Shorthand for <span class="math">\(\land\)</span> e.g. <span class="math">\((\forall x \in \{ 1, 2, ... 10 \}) x > 0 \equiv 1 > 0 \land 2 > 0 \land ... 10 > 0\)</span></li>
</ul></li>
<li><p><span class="math">\(\exists\)</span>:</p>
<ul>
<li>Shorthand for <span class="math">\(\lor\)</span> e.g. <span class="math">\((\exists x \in \{ 1, 2, ... 10 \}) x > 5 \equiv 1 > 5 \lor 2 > 5 \lor ... 10 > 5\)</span></li>
</ul></li>
<li><p><span class="math">\(\neg \forall \equiv \exists\)</span>, vice versa</p></li>
<li><p>With quantifiers we can write logical statements e.g.</p>
<ul>
<li><span class="math">\((\forall x \in \mathbb{R}) (\forall y \in \mathbb{R}) x > y\)</span> is a statement and is false</li>
<li><span class="math">\((\forall x) (\exists y) x > y\)</span> is true</li>
<li>shortcut: <span class="math">\((\exists x, y) \equiv (\exists x) (\exists y)\)</span></li>
</ul></li>
<li><p>Quantifiers can be expressed in set language, sort of a definition in terms of sets:</p>
<ul>
<li><span class="math">\((\forall x \in \mathbb{X}) (\phi(x)) \equiv \{ p \in \mathbb{X} : \phi(p) \} = \mathbb{X}\)</span></li>
<li><span class="math">\((\exists x \in \mathbb{X}) (\phi(x)) \equiv \{ q \in \mathbb{X} : \phi(q) \} \neq \emptyset\)</span></li>
<li><span class="math">\((\exists x \in \mathbb{X}) (\neg \phi(x)) \equiv \neg ( \{ p \in \mathbb{X} : \phi(p) \} = \mathbb{X} )\)</span></li>
</ul></li>
<li><p>Order of quantifiers matters.</p></li>
</ul>
<h2 id="relations">Relations</h2>
<ul>
<li><p>Cartesian product:</p>
<ul>
<li><span class="math">\(A \times B = \{ (p, q) : p \in A \land q \in B \}\)</span></li>
</ul></li>
<li><p>Def: A relation <span class="math">\(R\)</span> on a set <span class="math">\(\mathbb X\)</span> is a subset of <span class="math">\(\mathbb X \times \mathbb X\)</span> (<span class="math">\(R \subseteq \mathbb X \times \mathbb X\)</span>)</p></li>
<li><p>Graph of a function <span class="math">\(f()\)</span>: <span class="math">\(\{ (x, f(x) : x \in Dom(f) \}\)</span></p></li>
<li><p>Properties of:</p>
<ul>
<li>Reflexivity: <span class="math">\((\forall x \in \mathbb X ) (x, x) \in R \equiv (\forall x \in \mathbb X) x R x\)</span></li>
<li>Symmetricity: <span class="math">\([ (\forall x, y \in \mathbb X) (x, y) \in R \implies (y, x) \in R) ] \equiv [ (\forall x, y \in \mathbb X) ( x R y \implies y R x) ]\)</span></li>
<li>Transitivity: <span class="math">\((\forall x, y, z \in \mathbb X) (x R y \land y R z \implies x R z)\)</span></li>
<li>Antisymmetricity: <span class="math">\((\forall x, y \in \mathbb X) (x R y \land y R x \implies x = y)\)</span></li>
</ul></li>
<li><p>Equivalence relations:</p>
<ul>
<li>Def: <span class="math">\(R \subseteq \mathbb X \times \mathbb X\)</span> is said to be an equivalence relation <em>iff</em> <span class="math">\(R\)</span> is reflexive, symmetric and transitive.</li>
<li>Congruence modulo n: <span class="math">\(p R q \equiv n | p - q\)</span></li>
<li>Def R - and equivalence relation of <span class="math">\(\mathbb X\)</span>: The <em>equivalence class</em> of an element <span class="math">\(x \in \mathbb X\)</span> is the set <span class="math">\([x]_R = \{ y \in \mathbb X : x R y \}\)</span>
<ul>
<li>Every <span class="math">\(x \in \mathbb X\)</span> belongs to the equivalence class of some element <span class="math">\(a\)</span>.</li>
<li><span class="math">\((\forall x, y \in \mathbb X) ([x] \cap [y] \neq \emptyset \iff [x] = [y])\)</span></li>
</ul></li>
</ul></li>
<li><p>Partitions</p>
<ul>
<li><p>A partition is a set containing subsets of some set <span class="math">\(\mathbb X\)</span> such that their collective symmetric difference equals <span class="math">\(\mathbb X\)</span>. A partition of is a set <span class="math">\(\{ A_i: i \in \mathbb I \land A_i \subseteq \mathbb X \}\)</span> such that:</p>
<ul>
<li><span class="math">\((\forall x \in \mathbb X) (\exists j \in \mathbb I) (x \in A_j)\)</span></li>
<li><span class="math">\((\forall i, j \in \mathbb I) (i \neq j \implies A_i \cap A_j = \emptyset)\)</span></li>
</ul></li>
<li><p><span class="math">\(\{ A_i \}_{i \in \mathbb I}\)</span> is a partition <em>iff</em> there exists an equivalence relation <span class="math">\(R\)</span> on <span class="math">\(\mathbb X\)</span> such that:</p>
<ul>
<li><span class="math">\((\forall i \in \mathbb I) (\exists x \in \mathbb X) A_i = [x]_R\)</span></li>
<li><span class="math">\((\forall x \in \mathbb X) (\exists j \in \mathbb I) [x] = A_j\)</span></li>
</ul></li>
<li><p>The quotient set: <span class="math">\(\mathbb X / R = \{ [a] : a \in \mathbb X \}\)</span></p></li>
</ul></li>
</ul>
</div>
</div>
</div>
<footer class="footer">
<div class="footer__inner">
<div class="copyright copyright--user">
<span>© Abdulkadir Furkan Şanlı 2019 :: <a href="https://creativecommons.org/licenses/by-nd/4.0/">CC
BY-ND</a></span>
<span>:: theme by <a href="https://twitter.com/panr">panr</a></span>
</div>
</div>
</footer>
<script src="https://022385.xyz/assets/main.js"></script>
<script src="https://022385.xyz/assets/prism.js"></script>
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.11.1/dist/katex.min.css"
integrity="sha384-zB1R0rpPzHqg7Kpt0Aljp8JPLqbXI3bhnPWROx27a9N0Ll6ZP/+DiW/UqRcLbRjq" crossorigin="anonymous">
<script defer src="https://cdn.jsdelivr.net/npm/katex@0.11.1/dist/katex.min.js"
integrity="sha384-y23I5Q6l+B6vatafAwxRu/0oK/79VlbSz7Q9aiSZUvyWYIYsd+qj+o24G5ZU2zJz"
crossorigin="anonymous"></script>
<script defer src="https://cdn.jsdelivr.net/npm/katex@0.11.1/dist/contrib/auto-render.min.js"
integrity="sha384-kWPLUVMOks5AQFrykwIup5lo0m3iMkkHrD0uJ4H5cjeGihAutqP0yW0J6dpFiVkI" crossorigin="anonymous"
onload="renderMathInElement(document.body);"></script>
</div>
</body>
</html>