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<div class="container">
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<a href="/">
<div class="logo">
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@ -82,13 +84,21 @@
<li><a href="/about">About</a></li>
<ul class="menu__sub-inner">
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@ -131,10 +141,10 @@
<a href="https://022385.xyz/posts/eidma/">Introduction to Discrete Mathematics</a></h1>
<div class="post-meta">
<span class="post-date">
2019-03-26
2019-11-04
</span>
<span class="post-author">::
abdul</span>
Abdulkadir</span>
</div>
@ -142,8 +152,6 @@
#<a href="https://022385.xyz/tags/university-notes/">university-notes</a>&nbsp;
#<a href="https://022385.xyz/tags/"></a>&nbsp;
</span>
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<div class="post-content">
Discrete mathematics 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.
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.
</div>
@ -160,7 +170,7 @@
<div>
<a class="read-more button"
href="/posts/eidma/">Read more →</a>
href="/posts/eidma/">read more →</a>
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<span>© Abdulkadir Furkan Şanlı 2019 :: <a href="https://creativecommons.org/licenses/by-nd/4.0/">CC BY-ND</a></span>
<span>:: Theme made by <a href="https://twitter.com/panr">panr</a></span>
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<description> name: Abdulkadir Furkan Şanlı handle: abdulocracy contact: email: my handle at disroot dot org irc (freenode): abdulocracy </description>
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<title>Introduction to Discrete Mathematics</title>
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<description>Discrete mathematics 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>
<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." />
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2019-03-26
2019-11-04
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<span class="post-author">::
abdul
Abdulkadir
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#<a href="https://022385.xyz/tags/university-notes/">university-notes</a>&nbsp;
#<a href="https://022385.xyz/tags/"></a>&nbsp;
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<h1 id="discrete-mathematics">Discrete mathematics</h1>
<ul>
<ul>
<li>Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets.</li>
</ul>
@ -151,22 +163,27 @@
<ul>
<li>Comes from the linguistic concept that things can be either true or false.</li>
<li>We should avoid variables when forming statements, as they may change the logical value.
<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>In logic we do not use the equals sign, we use the equivalence sign <span class="math">\(\equiv\)</span>.</li>
<li>Logical values (booleans) are denoted by either 0 or 1 (or t, f, etc.).</li>
<li>When doing logic, we use propositional variables (e.g. p, q, r).
<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>The operations done on propositional variables are called propositional connectives.
<ol>
<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>)
@ -174,13 +191,15 @@
<ul>
<li><span class="math">\(\equiv \neg p \lor q\)</span></li>
</ul></li>
</ol></li>
<li>Not necessarily connectives but unary operations:
</ul></li>
<ol>
<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>
</ol></li>
<li>A (propositional) formula is a &quot;properly constructed&quot; logical expression.
</ul></li>
<li><p>A (propositional) formula is a &quot;properly constructed&quot; logical expression.</p>
<ul>
<li>e.g. <span class="math">\(\neg[(p \lor q)] \land r\)</span></li>
@ -195,7 +214,8 @@
<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>Double negation law: <span class="math">\(\neg(\neg p) \equiv p\)</span></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>
@ -204,13 +224,14 @@
<li><p>Contraposition law:</p>
<ul>
<li><span class="math">\((p \implies q) \equiv (\neg q \implies \neg p)\)</span> prove by contraposition
<li><p><span class="math">\((p \implies q) \equiv (\neg q \implies \neg p)\)</span> prove by contraposition</p>
<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>
<li>Contradiction law:
<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>
@ -223,15 +244,17 @@
<h2 id="sets">Sets</h2>
<ul>
<li>We will consider subsets of universal set <span class="math">\(\mathbb X\)</span>
<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><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></li>
<li>Operations on sets:
<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>
@ -242,25 +265,30 @@
<li>If <span class="math">\(x = \{ 1 \}\)</span> then <span class="math">\(x' = \emptyset\)</span></li>
</ul></li>
</ul></li>
<li>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></li>
<li>Difference of sets:
<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>Laws of set algebra:
<ol>
<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>
</ol></li>
<li>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>)</li>
<li>Quip: <span class="math">\(\{ x \in \mathbb R : x^2 = -1\} = \emptyset\)</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>
@ -268,45 +296,54 @@
<ul>
<li><span class="math">\(\phi\)</span> - prepositional function: yields only true or false value</li>
<li><span class="math">\(\forall\)</span> means &quot;for all&quot; and <span class="math">\(\exists\)</span> means &quot;there exists&quot;</li>
<li><span class="math">\(\forall\)</span>
<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>
<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><span class="math">\(\exists\)</span>
<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>
<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><span class="math">\(\neg \forall \equiv \exists\)</span>, vice versa</li>
<li>With quantifiers we can write logical statements e.g.
<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>Quantifiers can be expressed in set language, sort of a definition in terms of sets:
<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>Order of quantifiers matters.</li>
<li><p>Order of quantifiers matters.</p></li>
</ul>
<h2 id="relations">Relations</h2>
<ul>
<li>Cartesian product:
<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>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>)</li>
<li>Graph of a function <span class="math">\(f()\)</span>: <span class="math">\(\{ (x, f(x) : x \in Dom(f) \}\)</span></li>
<li>Properties of:
<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>
<ol>
<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>
@ -314,7 +351,8 @@
<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>
</ol></li>
<li>Equivalence relations:
<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>
@ -326,22 +364,25 @@
<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>Partitions
<li><p>Partitions</p>
<ul>
<li>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:
<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><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:
<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>The quotient set: <span class="math">\(\mathbb X / R = \{ [a] : a \in \mathbb X \}\)</span></li>
<li><p>The quotient set: <span class="math">\(\mathbb X / R = \{ [a] : a \in \mathbb X \}\)</span></p></li>
</ul></li>
</ul>
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<a href="https://022385.xyz/posts/eidma/">Introduction to Discrete Mathematics</a></h1>
<div class="post-meta">
<span class="post-date">
2019-03-26
2019-11-04
</span>
<span class="post-author">::
abdul</span>
Abdulkadir</span>
</div>
@ -139,8 +149,6 @@
#<a href="https://022385.xyz/tags/university-notes/">university-notes</a>&nbsp;
#<a href="https://022385.xyz/tags/"></a>&nbsp;
</span>
@ -149,7 +157,9 @@
<div class="post-content">
Discrete mathematics 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.
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.
</div>
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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 \(\equiv\).
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<a href="https://022385.xyz/posts/eidma/">Introduction to Discrete Mathematics</a></h1>
<div class="post-meta">
<span class="post-date">
2019-03-26
2019-11-04
</span>
<span class="post-author">::
abdul</span>
Abdulkadir</span>
</div>
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#<a href="https://022385.xyz/tags/university-notes/">university-notes</a>&nbsp;
#<a href="https://022385.xyz/tags/"></a>&nbsp;
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Discrete mathematics 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.
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.
</div>
@ -157,7 +167,7 @@
<div>
<a class="read-more button"
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href="/posts/eidma/">read more →</a>
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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.
\(2=7\) 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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