452 lines
24 KiB
HTML
452 lines
24 KiB
HTML
<!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://abdulocra.cy/posts/eidma/" />
|
||
|
||
|
||
<link rel="stylesheet" href="https://abdulocra.cy/assets/style.css">
|
||
|
||
<link rel="stylesheet" href="https://abdulocra.cy/assets/pink.css">
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<link rel="apple-touch-icon-precomposed" sizes="144x144" href="https://abdulocra.cy/img/apple-touch-icon-144-precomposed.png">
|
||
|
||
<link rel="shortcut icon" href="https://abdulocra.cy/img/favicon/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://abdulocra.cy/" />
|
||
<meta name="twitter:creator" content="" />
|
||
<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://abdulocra.cy/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="2019-11-20 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">menu ▾</li>
|
||
|
||
<ul class="menu__sub-inner-more hidden">
|
||
|
||
|
||
<li><a href="/about">about</a></li>
|
||
|
||
|
||
|
||
<li><a href="https://music.abdulocra.cy">music</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="https://music.abdulocra.cy">music</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://abdulocra.cy/posts/eidma/">Introduction to Discrete Mathematics</a></h1>
|
||
<div class="post-meta">
|
||
|
||
<span class="post-date">
|
||
2019-11-20
|
||
</span>
|
||
|
||
|
||
</div>
|
||
|
||
|
||
<span class="post-tags">
|
||
|
||
#<a href="https://abdulocra.cy/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><p>Comes from the linguistic concept that things can be either true or false.</p></li>
|
||
<li><p>We should avoid variables when forming statements, as they may change the logical value.</p>
|
||
<ul>
|
||
<li><span class="math inline">\(2=7\)</span> statement</li>
|
||
<li><span class="math inline">\(x=5\)</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 inline">\(\equiv\)</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 inline">\(p \land q\)</span> is only true if both p and q are true <span class="math inline">\((0001)\)</span></li>
|
||
<li>Disjunction: <span class="math inline">\(p \lor q\)</span> is only false if both p and q are false <span class="math inline">\((0111)\)</span></li>
|
||
<li>Implication (material conditional): <span class="math inline">\(p \implies q\)</span> is false only if p is true and q is false (truth table <span class="math inline">\((1011)\)</span>)
|
||
<ul>
|
||
<li><span class="math inline">\(\equiv \neg p \lor q\)</span></li>
|
||
</ul></li>
|
||
</ul></li>
|
||
<li><p>Not necessarily connectives but unary operations:</p>
|
||
<ul>
|
||
<li>Negation: Denoted by ~, <span class="math inline">\(\neg\)</span> or NOT, negates the one input <span class="math inline">\((10)\)</span>.</li>
|
||
</ul></li>
|
||
<li><p>A (propositional) formula is a “properly constructed” logical expression.</p>
|
||
<ul>
|
||
<li>e.g. <span class="math inline">\(\neg[(p \lor q)] \land r\)</span></li>
|
||
<li><span class="math inline">\((p \land)\)</span> is not a formula, as <span class="math inline">\(\land\)</span> requires 2 variables.</li>
|
||
<li>Logical equivalence: <span class="math inline">\(\phi(p, q, k) \equiv \psi(p, q, k)\)</span>, logical value of <span class="math inline">\(\phi\)</span> is equal to logical value of <span class="math inline">\(\psi\)</span>.</li>
|
||
<li>Commutativity: <span class="math inline">\(p \land q \equiv q \land p\)</span></li>
|
||
<li>Associativity: <span class="math inline">\((p \land q) \land r \equiv p \land (q \land r)\)</span></li>
|
||
<li>Distributivity: <span class="math inline">\(p \land (q \lor r) \equiv (p \land q) \lor (p \land r)\)</span></li>
|
||
<li>Conjunctive normal form: every formula can be written as a conjunction of one or more disjunctions.
|
||
<ul>
|
||
<li><span class="math inline">\(\neg(B \lor C)\)</span> can be written as <span class="math inline">\(\neg B \land \neg C\)</span></li>
|
||
</ul></li>
|
||
</ul></li>
|
||
<li><p>Double negation law: <span class="math inline">\(\neg(\neg p) \equiv p\)</span></p></li>
|
||
<li><p>De Morgan’s laws: <span class="math inline">\(\neg(p \land q) \equiv \neg p \lor \neg q\)</span> and <span class="math inline">\(\neg(p \lor q) \equiv \neg p \land \neg q\)</span>.</p></li>
|
||
<li><p>If and only if (<em>iff</em>): <span class="math inline">\(p \iff p \equiv (p \implies q) \land (q \implies p)\)</span></p></li>
|
||
<li><p>Contraposition law:</p>
|
||
<ul>
|
||
<li><span class="math inline">\((p \implies q) \equiv (\neg q \implies \neg p)\)</span> prove by contraposition
|
||
<ul>
|
||
<li><span class="math inline">\((p \implies q) \equiv (\neg p \lor q)\)</span></li>
|
||
<li><span class="math inline">\((\neg q \implies \neg p) \equiv (\neg (\neg q) \lor (\neg p) \equiv (q \lor \neg p) \equiv (\neg p \lor q)\)</span></li>
|
||
</ul></li>
|
||
</ul></li>
|
||
<li><p>Contradiction law:</p>
|
||
<ul>
|
||
<li><span class="math inline">\(p \lor \neg p \equiv 1\)</span> and <span class="math inline">\(p \land \neg p \equiv 0\)</span></li>
|
||
</ul></li>
|
||
<li><p>Tautology: <span class="math inline">\(\phi (p, q, ... r)\)</span> is a tautology <em>iff</em> <span class="math inline">\(\phi \equiv 1\)</span></p></li>
|
||
</ul>
|
||
<h2 id="sets">Sets</h2>
|
||
<ul>
|
||
<li>We will consider subsets of universal set <span class="math inline">\(\mathbb X\)</span>
|
||
<ul>
|
||
<li><span class="math inline">\(2^\mathbb X = \{ A : A \subseteq \mathbb X\}\)</span></li>
|
||
<li><span class="math inline">\(2^\mathbb X = P(\mathbb X)\)</span></li>
|
||
<li>All 2 object subsets of <span class="math inline">\(\mathbb X\)</span>: <span class="math inline">\(P_2(\mathbb X)\)</span></li>
|
||
</ul></li>
|
||
<li><span class="math inline">\(A \subset B \equiv\)</span> every element of A is an element of B <span class="math inline">\(\equiv \{x \in \mathbb X : x \in A \implies x \in B\}\)</span></li>
|
||
<li>Operations on sets:
|
||
<ul>
|
||
<li>Union - <span class="math inline">\(\cup\)</span> - <span class="math inline">\(A \cup B = \{ x \in \mathbb X : x \in A \lor x \in B \}\)</span></li>
|
||
<li>Intersection - <span class="math inline">\(\cap\)</span> - <span class="math inline">\(A \cap B = \{ x \in \mathbb X : x \in A \land x \in B \}\)</span></li>
|
||
<li>Complement - <span class="math inline">\(A'\)</span> - <span class="math inline">\(A' = \{ x \in \mathbb X : \neg (x \in A) \}\)</span>
|
||
<ul>
|
||
<li>If <span class="math inline">\(x = \{ 1 \}\)</span> then <span class="math inline">\(x' = \emptyset\)</span></li>
|
||
</ul></li>
|
||
</ul></li>
|
||
<li>Equality of sets: <span class="math inline">\(A = B\)</span> iff <span class="math inline">\(x \in \mathbb X : (x \in A \iff x \in B)\)</span></li>
|
||
<li>Difference of sets:
|
||
<ul>
|
||
<li><span class="math inline">\(A \setminus B = \{ x \in \mathbb X : x \in A \land x \notin B \} = A \cap B'\)</span></li>
|
||
<li>Symmetric difference: <span class="math inline">\(A \div B = (A \setminus B) \cup (B \setminus A)\)</span></li>
|
||
</ul></li>
|
||
<li>Laws of set algebra:
|
||
<ul>
|
||
<li><span class="math inline">\(A \cup B = B \cup A , A \cap B = B \cap A\)</span></li>
|
||
<li><span class="math inline">\((A \cup B) \cup C = A \cup (B \cup C), (A \cap B) \cap C = A \cap (B \cap C)\)</span></li>
|
||
<li><span class="math inline">\((A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\)</span> vice versa</li>
|
||
<li><span class="math inline">\(A \cap \emptyset, A \cap \mathbb X = A, A \cup \emptyset = A, A \cup \mathbb X = \mathbb X\)</span></li>
|
||
<li><span class="math inline">\((A \cup B)' = A' \cap B'\)</span> vice versa</li>
|
||
<li><span class="math inline">\(A \cup A' = \mathbb X, A \cap A' = \emptyset\)</span></li>
|
||
</ul></li>
|
||
<li>Note: <span class="math inline">\(\{ \emptyset \} \neq \emptyset\)</span>, one is a set with one element, one is the empty set, no elements (<span class="math inline">\(\{ \}\)</span>)</li>
|
||
<li>Quip: <span class="math inline">\(\{ x \in \mathbb R : x^2 = -1\} = \emptyset\)</span></li>
|
||
</ul>
|
||
<h2 id="quantifiers">Quantifiers</h2>
|
||
<ul>
|
||
<li><span class="math inline">\(\phi\)</span> - prepositional function: yields only true or false value</li>
|
||
<li><span class="math inline">\(\forall\)</span> means “for all” and <span class="math inline">\(\exists\)</span> means “there exists”</li>
|
||
<li><span class="math inline">\(\forall\)</span>:
|
||
<ul>
|
||
<li>Shorthand for <span class="math inline">\(\land\)</span> e.g. <span class="math inline">\((\forall x \in \{ 1, 2, ... 10 \}) x > 0 \equiv 1 > 0 \land 2 > 0 \land ... 10 > 0\)</span></li>
|
||
</ul></li>
|
||
<li><span class="math inline">\(\exists\)</span>:
|
||
<ul>
|
||
<li>Shorthand for <span class="math inline">\(\lor\)</span> e.g. <span class="math inline">\((\exists x \in \{ 1, 2, ... 10 \}) x > 5 \equiv 1 > 5 \lor 2 > 5 \lor ... 10 > 5\)</span></li>
|
||
</ul></li>
|
||
<li><span class="math inline">\(\neg \forall \equiv \exists\)</span>, vice versa</li>
|
||
<li>With quantifiers we can write logical statements e.g.
|
||
<ul>
|
||
<li><span class="math inline">\((\forall x \in \mathbb{R}) (\forall y \in \mathbb{R}) x > y\)</span> is a statement and is false</li>
|
||
<li><span class="math inline">\((\forall x) (\exists y) x > y\)</span> is true</li>
|
||
<li>shortcut: <span class="math inline">\((\exists x, y) \equiv (\exists x) (\exists y)\)</span></li>
|
||
</ul></li>
|
||
<li>Quantifiers can be expressed in set language, sort of a definition in terms of sets:
|
||
<ul>
|
||
<li><span class="math inline">\((\forall x \in \mathbb{X}) (\phi(x)) \equiv \{ p \in \mathbb{X} : \phi(p) \} = \mathbb{X}\)</span></li>
|
||
<li><span class="math inline">\((\exists x \in \mathbb{X}) (\phi(x)) \equiv \{ q \in \mathbb{X} : \phi(q) \} \neq \emptyset\)</span></li>
|
||
<li><span class="math inline">\((\exists x \in \mathbb{X}) (\neg \phi(x)) \equiv \neg ( \{ p \in \mathbb{X} : \phi(p) \} = \mathbb{X} )\)</span></li>
|
||
</ul></li>
|
||
<li>Order of quantifiers matters.</li>
|
||
</ul>
|
||
<h2 id="relations">Relations</h2>
|
||
<ul>
|
||
<li>Cartesian product:
|
||
<ul>
|
||
<li><span class="math inline">\(A \times B = \{ (p, q) : p \in A \land q \in B \}\)</span></li>
|
||
</ul></li>
|
||
<li>Def: A relation <span class="math inline">\(R\)</span> on a set <span class="math inline">\(\mathbb X\)</span> is a subset of <span class="math inline">\(\mathbb X \times \mathbb X\)</span> (<span class="math inline">\(R \subseteq \mathbb X \times \mathbb X\)</span>)</li>
|
||
<li>Graph of a function <span class="math inline">\(f()\)</span>: <span class="math inline">\(\{ (x, f(x) : x \in Dom(f) \}\)</span></li>
|
||
<li>Properties of:
|
||
<ul>
|
||
<li>Reflexivity: <span class="math inline">\((\forall x \in \mathbb X ) (x, x) \in R \equiv (\forall x \in \mathbb X) x R x\)</span></li>
|
||
<li>Symmetricity: <span class="math inline">\([ (\forall x, y \in \mathbb X) (x, y) \in R \implies (y, x) \in R) ] \equiv [ (\forall x, y \in \mathbb X) ( x R y \implies y R x) ]\)</span></li>
|
||
<li>Transitivity: <span class="math inline">\((\forall x, y, z \in \mathbb X) (x R y \land y R z \implies x R z)\)</span></li>
|
||
<li>Antisymmetricity: <span class="math inline">\((\forall x, y \in \mathbb X) (x R y \land y R x \implies x = y)\)</span></li>
|
||
</ul></li>
|
||
<li>Equivalence relations:
|
||
<ul>
|
||
<li>Def: <span class="math inline">\(R \subseteq \mathbb X \times \mathbb X\)</span> is said to be an equivalence relation <em>iff</em> <span class="math inline">\(R\)</span> is reflexive, symmetric and transitive.</li>
|
||
<li>Congruence modulo n: <span class="math inline">\(p R q \equiv n | p - q\)</span></li>
|
||
<li>Def R - and equivalence relation of <span class="math inline">\(\mathbb X\)</span>: The <em>equivalence class</em> of an element <span class="math inline">\(x \in \mathbb X\)</span> is the set <span class="math inline">\([x]_R = \{ y \in \mathbb X : x R y \}\)</span>
|
||
<ul>
|
||
<li>Every <span class="math inline">\(x \in \mathbb X\)</span> belongs to the equivalence class of some element <span class="math inline">\(a\)</span>.</li>
|
||
<li><span class="math inline">\((\forall x, y \in \mathbb X) ([x] \cap [y] \neq \emptyset \iff [x] = [y])\)</span></li>
|
||
</ul></li>
|
||
</ul></li>
|
||
<li>Partitions
|
||
<ul>
|
||
<li>A partition is a set containing subsets of some set <span class="math inline">\(\mathbb X\)</span> such that their collective symmetric difference equals <span class="math inline">\(\mathbb X\)</span>. A partition of is a set <span class="math inline">\(\{ A_i: i \in \mathbb I \land A_i \subseteq \mathbb X \}\)</span> such that:
|
||
<ul>
|
||
<li><span class="math inline">\((\forall x \in \mathbb X) (\exists j \in \mathbb I) (x \in A_j)\)</span></li>
|
||
<li><span class="math inline">\((\forall i, j \in \mathbb I) (i \neq j \implies A_i \cap A_j = \emptyset)\)</span></li>
|
||
</ul></li>
|
||
<li><span class="math inline">\(\{ A_i \}_{i \in \mathbb I}\)</span> is a partition <em>iff</em> there exists an equivalence relation <span class="math inline">\(R\)</span> on <span class="math inline">\(\mathbb X\)</span> such that:
|
||
<ul>
|
||
<li><span class="math inline">\((\forall i \in \mathbb I) (\exists x \in \mathbb X) A_i = [x]_R\)</span></li>
|
||
<li><span class="math inline">\((\forall x \in \mathbb X) (\exists j \in \mathbb I) [x] = A_j\)</span></li>
|
||
</ul></li>
|
||
<li>The quotient set: <span class="math inline">\(\mathbb X / R = \{ [a] : a \in \mathbb X \}\)</span></li>
|
||
</ul></li>
|
||
</ul>
|
||
<h2 id="posets">Posets</h2>
|
||
<ul>
|
||
<li>Partial orders
|
||
<ul>
|
||
<li><span class="math inline">\(\mathbb X\)</span> is a set, <span class="math inline">\(R \subseteq \mathbb X \times \mathbb X\)</span></li>
|
||
<li>Def: <span class="math inline">\(R\)</span> is a partial order on <span class="math inline">\(\mathbb X\)</span> iff <span class="math inline">\(R\)</span> is:
|
||
<ul>
|
||
<li>Reflexive</li>
|
||
<li>Antisymmetric</li>
|
||
<li>Transitive</li>
|
||
</ul></li>
|
||
<li>Def: <span class="math inline">\(m \in \mathbb X\)</span> is said to be:
|
||
<ul>
|
||
<li>maximal element in <span class="math inline">\((\mathbb X, \preccurlyeq)\)</span> iff <span class="math inline">\((\forall a \in \mathbb X) m \preccurlyeq a \implies m = a\)</span></li>
|
||
<li>largest iff <span class="math inline">\((\forall a \in \mathbb X) (a \preccurlyeq m)\)</span></li>
|
||
<li>minimal iff <span class="math inline">\((\forall a \in \mathbb X) (a \preccurlyeq m \implies a = m)\)</span></li>
|
||
<li>smallest iff <span class="math inline">\((\forall a \in \mathbb X) (m \preccurlyeq a)\)</span></li>
|
||
</ul></li>
|
||
<li>Def: A partial order <span class="math inline">\(R\)</span> on <span class="math inline">\(\mathbb X\)</span> is said to be <em>“total”</em> iff <span class="math inline">\((\forall x, y \in \mathbb X) (x R y \lor y R x)\)</span></li>
|
||
<li>Def: A subset <span class="math inline">\(B\)</span> of <span class="math inline">\(\mathbb X\)</span> is called a chain <em>“chain”</em> iff <span class="math inline">\(B\)</span> is totally ordered by <span class="math inline">\(R\)</span>
|
||
<ul>
|
||
<li><span class="math inline">\(C(\mathbb X)\)</span> - the set of all chains in <span class="math inline">\((\mathbb X, R)\)</span></li>
|
||
<li>A chain <span class="math inline">\(D\)</span> in <span class="math inline">\((\mathbb X, R)\)</span> is called a maximal chain iff <span class="math inline">\(D\)</span> is a maximal element in <span class="math inline">\((C(\mathbb X), R)\)</span></li>
|
||
<li><span class="math inline">\(K \subseteq \mathbb X\)</span> is called an antichain in <span class="math inline">\((\mathbb X, R)\)</span> iff <span class="math inline">\((\forall p, q \in K) (p R q \lor q R p \implies p = q)\)</span></li>
|
||
<li>Def: <span class="math inline">\(R\)</span> is a partial order on <span class="math inline">\(\mathbb X\)</span>, <span class="math inline">\(R\)</span> is called a <em>well</em> order iff <span class="math inline">\(R\)</span> is a total order on <span class="math inline">\(X\)</span> and every nonempty subset <span class="math inline">\(A\)</span> of <span class="math inline">\(\mathbb X\)</span> has the smallest element</li>
|
||
</ul></li>
|
||
</ul></li>
|
||
</ul>
|
||
<h2 id="induction">Induction</h2>
|
||
<ul>
|
||
<li>If <span class="math inline">\(\phi\)</span> is a propositional function defined on <span class="math inline">\(\mathbb N\)</span>, if:
|
||
<ul>
|
||
<li><span class="math inline">\(\phi(1)\)</span></li>
|
||
<li><span class="math inline">\((\forall n \geq 1) \phi(n) \implies \phi(n+1)\)</span></li>
|
||
<li><span class="math inline">\((\forall k \geq 1) \phi(k)\)</span></li>
|
||
</ul></li>
|
||
</ul>
|
||
<h2 id="functions">Functions</h2>
|
||
<ul>
|
||
<li><span class="math inline">\(f: \mathbb X \to \mathbb Y\)</span></li>
|
||
<li>Def: <span class="math inline">\(f \subseteq \mathbb X \times \mathbb Y\)</span> is said to be a function if:
|
||
<ul>
|
||
<li><span class="math inline">\((\forall x \in \mathbb X)(\exists y \in \mathbb Y) (x, y) \in f(y = f(x))\)</span></li>
|
||
<li><span class="math inline">\((\forall a \in \mathbb X)(\forall p, q \in \mathbb Y)((a, p) \in f \land (a, q) \in f \implies p = q)\)</span></li>
|
||
</ul></li>
|
||
<li>Types of functions <span class="math inline">\(f: \mathbb X \to \mathbb Y\)</span>:
|
||
<ul>
|
||
<li><span class="math inline">\(f\)</span> is said to be an injection ( 1 to 1 function) iff <span class="math inline">\((\forall x_1, x_2 \in \mathbb X) x_1 \neq x_2 \implies f(x_1) \neq f(x_2)\)</span></li>
|
||
<li><span class="math inline">\(f\)</span> is said to be a surjection (onto function) iff <span class="math inline">\((\forall y \in \mathbb Y)(\exists x \in \mathbb X) f(x) = y\)</span></li>
|
||
<li>If <span class="math inline">\(f^{-1}\)</span> is a function from <span class="math inline">\(\mathbb Y \to \mathbb X\)</span> then <span class="math inline">\(f^{-1}\)</span> is called the inverse function for <span class="math inline">\(f\)</span>
|
||
<ul>
|
||
<li>Fact: <span class="math inline">\(f^{-1}\)</span> is a function iff <span class="math inline">\(f\)</span> is a <em>bijection</em> (1 to 1 and onto)</li>
|
||
</ul></li>
|
||
</ul></li>
|
||
<li>For some set <span class="math inline">\(\mathbb A\)</span> the image of <span class="math inline">\(\mathbb A\)</span> by <span class="math inline">\(f\)</span> is <span class="math inline">\(f(\mathbb A) = \{ f(x) : x \in \mathbb A \}\)</span>. We can also define the inverse of an image even when the function itself isn’t invertible: <span class="math inline">\(f^{-1}(\mathbb A)\)</span></li>
|
||
</ul>
|
||
<h2 id="combinatorics">Combinatorics</h2>
|
||
<ul>
|
||
<li><span class="math inline">\(|\mathbb A|\)</span> size (number of elements) of <span class="math inline">\(\mathbb A\)</span></li>
|
||
<li>Rule of addition:
|
||
<ul>
|
||
<li>If <span class="math inline">\(\mathbb A, \mathbb B \subseteq \mathbb X\)</span> and <span class="math inline">\(|\mathbb A|, |\mathbb B| \in \mathbb N\)</span> and <span class="math inline">\(\mathbb A \cap \mathbb B = \emptyset\)</span> then <span class="math inline">\(|\mathbb A \cup \mathbb B| = |\mathbb A| + |\mathbb B|\)</span></li>
|
||
<li>Can be generalized as: <span class="math display">\[
|
||
(\forall n ) \mathbb{A}_1, \mathbb{A}_2, ..., \mathbb{A}_n \in \mathbb{X} \land \\
|
||
|\mathbb{A}_1|, |\mathbb{A}_2|, ..., |\mathbb{A}_n| \in \mathbb{N} \implies \\
|
||
(\forall i, j \in \{1, 2, ..., n \})(i \neq j \implies \mathbb{A}_i \cap \mathbb{A}_j = \emptyset)
|
||
\]</span></li>
|
||
</ul></li>
|
||
<li>Rule of multiplication:
|
||
<ul>
|
||
<li><span class="math inline">\(\mathbb{A}, \mathbb{B} \subseteq \mathbb{X}, |\mathbb{A} \times \mathbb{B}| = |\mathbb{A}| \cdot |\mathbb{B}|\)</span></li>
|
||
<li>Can be generalized as: <span class="math display">\[
|
||
(\forall n ) \mathbb{A}_1, \mathbb{A}_2, ..., \mathbb{A}_n \in \mathbb{X} \land |\mathbb{A}_i| \in \mathbb{N} \implies \\
|
||
|\mathbb{A}_1 \times \mathbb{A}_2 \times ... \times \mathbb{A}_n| = |\mathbb{A}_1| \cdot |\mathbb{A}_2| \cdot ... \cdot |\mathbb{A_n}|
|
||
\]</span></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://abdulocra.cy/assets/main.js"></script>
|
||
<script src="https://abdulocra.cy/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>
|