Switch back to KaTeX, with Pandoc markup

Signed-off-by: Abdulkadir Furkan Şanlı <abdulkadirfsanli@protonmail.com>
This commit is contained in:
Abdulkadir Furkan Şanlı 2019-11-21 20:51:06 +01:00
parent d5e78f2838
commit 835828a38e
No known key found for this signature in database
GPG Key ID: 7823BD18E6F95D73
12 changed files with 198 additions and 311 deletions

View File

@ -2,6 +2,7 @@
title = "Introduction to Discrete Mathematics"
date = "2019-11-20"
tags = ["university-notes"]
markup = "pandoc"
+++
- Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets.

View File

@ -3,7 +3,7 @@ To add an extended footer section, please create
`layouts/partials/extended_footer.html` in your Hugo directory.
-->
<!-- KaTeX
<!-- KaTeX -->
<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"
@ -12,8 +12,8 @@ To add an extended footer section, please create
<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>
-->
<!-- MathJax
<script>
MathJax = {
tex: {
@ -24,3 +24,4 @@ To add an extended footer section, please create
</script>
<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js">
</script>
-->

View File

@ -181,17 +181,16 @@
<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>
<script>
MathJax = {
tex: {
inlineMath: [['$', '$'], ['\\(', '\\)']],
displayMath: [['$$', '$$'], ['\[', '\]']]
}
};
</script>
<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js">
</script>

View File

@ -164,17 +164,16 @@
<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>
<script>
MathJax = {
tex: {
inlineMath: [['$', '$'], ['\\(', '\\)']],
displayMath: [['$$', '$$'], ['\[', '\]']]
}
};
</script>
<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js">
</script>

View File

@ -159,9 +159,7 @@
<div class="post-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.
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>
@ -205,17 +203,16 @@
<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>
<script>
MathJax = {
tex: {
inlineMath: [['$', '$'], ['\\(', '\\)']],
displayMath: [['$$', '$$'], ['\[', '\]']]
}
};
</script>
<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js">
</script>

View File

@ -18,9 +18,7 @@
<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>
<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>

View File

@ -6,9 +6,7 @@
<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="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/" />
@ -31,9 +29,7 @@
<meta name="twitter:card" content="summary" />
<meta name="twitter:title" content="Introduction to Discrete Mathematics :: abdulocracy&#39;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: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="">
@ -42,9 +38,7 @@
<meta property="og:locale" content="en" />
<meta property="og:type" content="article" />
<meta property="og:title" content="Introduction to Discrete Mathematics :: abdulocracy&#39;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: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="">
@ -151,314 +145,224 @@
<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>
<li>We should avoid variables when forming statements, as they may change the logical value.
<ul>
<li>$2=7$ statement</li>
<li>$x=5$ not a statement</li>
<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 $\equiv$.</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>
<li>In logic we do not use the equals sign, we use the equivalence sign <span class="math inline">\(\equiv\)</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).
<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>
<li>The operations done on propositional variables are called propositional connectives.
<ul>
<li>Conjunction: $p \land q$ is only true if both p and q are true $(0001)$</li>
<li>Disjunction: $p \lor q$ is only false if both p and q are false $(0111)$</li>
<li>Implication (material conditional): $p \implies q$ is false only if p is true and q is false (truth table $(1011)$)
<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>$\equiv \neg p \lor q$</li>
<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>
<li>Not necessarily connectives but unary operations:
<ul>
<li>Negation: Denoted by ~, $\neg$ or NOT, negates the one input $(10)$.</li>
<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 &ldquo;properly constructed&rdquo; logical expression.</p>
<li>A (propositional) formula is a “properly constructed” logical expression.
<ul>
<li>e.g. $\neg[(p \lor q)] \land r$</li>
<li>$(p \land)$ is not a formula, as $\land$ requires 2 variables.</li>
<li>Logical equivalence: $\phi(p, q, k) \equiv \psi(p, q, k)$, logical value of $\phi$ is equal to logical value of $\psi$.</li>
<li>Commutativity: $p \land q \equiv q \land p$</li>
<li>Associativity: $(p \land q) \land r \equiv p \land (q \land r)$</li>
<li>Distributivity: $p \land (q \lor r) \equiv (p \land q) \lor (p \land r)$</li>
<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>$\neg(B \lor C)$ can be written as $\neg B \land \neg C$</li>
<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: $\neg(\neg p) \equiv p$</p></li>
<li><p>De Morgan&rsquo;s laws: $\neg(p \land q) \equiv \neg p \lor \neg q$ and $\neg(p \lor q) \equiv \neg p \land \neg q$.</p></li>
<li><p>If and only if (<em>iff</em>): $p \iff p \equiv (p \implies q) \land (q \implies p)$</p></li>
<li><p>Contraposition law:</p>
<li>Double negation law: <span class="math inline">\(\neg(\neg p) \equiv p\)</span></li>
<li><p>De Morgans 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>If and only if (<em>iff</em>): <span class="math inline">\(p \iff p \equiv (p \implies q) \land (q \implies p)\)</span></li>
<li>Contraposition law:
<ul>
<li>$(p \implies q) \equiv (\neg q \implies \neg p)$ prove by contraposition
<li><span class="math inline">\((p \implies q) \equiv (\neg q \implies \neg p)\)</span> prove by contraposition
<ul>
<li>$(p \implies q) \equiv (\neg p \lor q)$</li>
<li>$(\neg q \implies \neg p) \equiv (\neg (\neg q) \lor (\neg p) \equiv (q \lor \neg p) \equiv (\neg p \lor q)$</li>
<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>
<li>Contradiction law:
<ul>
<li>$p \lor \neg p \equiv 1$ and $p \land \neg p \equiv 0$</li>
<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: $\phi (p, q, &hellip; r)$ is a tautology <em>iff</em> $\phi \equiv 1$</p></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><p>We will consider subsets of universal set $\mathbb X$</p>
<li>We will consider subsets of universal set <span class="math inline">\(\mathbb X\)</span>
<ul>
<li>$2^\mathbb X = { A : A \subseteq \mathbb X}$</li>
<li>$2^\mathbb X = P(\mathbb X)$</li>
<li>All 2 object subsets of $\mathbb X$: $P_2(\mathbb X)$</li>
<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><p>$A \subset B \equiv$ every element of A is an element of B $\equiv {x \in \mathbb X : x \in A \implies x \in B}$</p></li>
<li><p>Operations on sets:</p>
<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 - $\cup$ - $A \cup B = { x \in \mathbb X : x \in A \lor x \in B }$</li>
<li>Intersection - $\cap$ - $A \cap B = { x \in \mathbb X : x \in A \land x \in B }$</li>
<li>Complement - $A&rsquo;$ - $A&rsquo; = { x \in \mathbb X : \neg (x \in A) }$
<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">\(A&#39;\)</span> - <span class="math inline">\(A&#39; = \{ x \in \mathbb X : \neg (x \in A) \}\)</span>
<ul>
<li>If $x = { 1 }$ then $x&rsquo; = \emptyset$</li>
<li>If <span class="math inline">\(x = \{ 1 \}\)</span> then <span class="math inline">\(x&#39; = \emptyset\)</span></li>
</ul></li>
</ul></li>
<li><p>Equality of sets: $A = B$ iff $x \in \mathbb X : (x \in A \iff x \in B)$</p></li>
<li><p>Difference of sets:</p>
<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>$A \setminus B = { x \in \mathbb X : x \in A \land x \notin B } = A \cap B&rsquo;$</li>
<li>Symmetric difference: $A \div B = (A \setminus B) \cup (B \setminus A)$</li>
<li><span class="math inline">\(A \setminus B = \{ x \in \mathbb X : x \in A \land x \notin B \} = A \cap B&#39;\)</span></li>
<li>Symmetric difference: <span class="math inline">\(A \div B = (A \setminus B) \cup (B \setminus A)\)</span></li>
</ul></li>
<li><p>Laws of set algebra:</p>
<li>Laws of set algebra:
<ul>
<li>$A \cup B = B \cup A , A \cap B = B \cap A$</li>
<li>$(A \cup B) \cup C = A \cup (B \cup C), (A \cap B) \cap C = A \cap (B \cap C)$</li>
<li>$(A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ vice versa</li>
<li>$A \cap \emptyset, A \cap \mathbb X = A, A \cup \emptyset = A, A \cup \mathbb X = \mathbb X$</li>
<li>$(A \cup B)&rsquo; = A&rsquo; \cap B&rsquo;$ vice versa</li>
<li>$A \cup A&rsquo; = \mathbb X, A \cap A&rsquo; = \emptyset$</li>
<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">\((A \cup B)&#39; = A&#39; \cap B&#39;\)</span> vice versa</li>
<li><span class="math inline">\(A \cup A&#39; = \mathbb X, A \cap A&#39; = \emptyset\)</span></li>
</ul></li>
<li><p>Note: ${ \emptyset } \neq \emptyset$, one is a set with one element, one is the empty set, no elements (${ }$)</p></li>
<li><p>Quip: ${ x \in \mathbb R : x^2 = -1} = \emptyset$</p></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>$\phi$ - prepositional function: yields only true or false value</li>
<li>$\forall$ means &ldquo;for all&rdquo; and $\exists$ means &ldquo;there exists&rdquo;</li>
<li><p>$\forall$:</p>
<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 $\land$ e.g. $(\forall x \in { 1, 2, &hellip; 10 }) x &gt; 0 \equiv 1 &gt; 0 \land 2 &gt; 0 \land &hellip; 10 &gt; 0$</li>
<li>Shorthand for <span class="math inline">\(\land\)</span> e.g. <span class="math inline">\((\forall x \in \{ 1, 2, ... 10 \}) x &gt; 0 \equiv 1 &gt; 0 \land 2 &gt; 0 \land ... 10 &gt; 0\)</span></li>
</ul></li>
<li><p>$\exists$:</p>
<li><span class="math inline">\(\exists\)</span>:
<ul>
<li>Shorthand for $\lor$ e.g. $(\exists x \in { 1, 2, &hellip; 10 }) x &gt; 5 \equiv 1 &gt; 5 \lor 2 &gt; 5 \lor &hellip; 10 &gt; 5$</li>
<li>Shorthand for <span class="math inline">\(\lor\)</span> e.g. <span class="math inline">\((\exists x \in \{ 1, 2, ... 10 \}) x &gt; 5 \equiv 1 &gt; 5 \lor 2 &gt; 5 \lor ... 10 &gt; 5\)</span></li>
</ul></li>
<li><p>$\neg \forall \equiv \exists$, vice versa</p></li>
<li><p>With quantifiers we can write logical statements e.g.</p>
<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>$(\forall x \in \mathbb{R}) (\forall y \in \mathbb{R}) x &gt; y$ is a statement and is false</li>
<li>$(\forall x) (\exists y) x &gt; y$ is true</li>
<li>shortcut: $(\exists x, y) \equiv (\exists x) (\exists y)$</li>
<li><span class="math inline">\((\forall x \in \mathbb{R}) (\forall y \in \mathbb{R}) x &gt; y\)</span> is a statement and is false</li>
<li><span class="math inline">\((\forall x) (\exists y) x &gt; y\)</span> is true</li>
<li>shortcut: <span class="math inline">\((\exists x, y) \equiv (\exists x) (\exists y)\)</span></li>
</ul></li>
<li><p>Quantifiers can be expressed in set language, sort of a definition in terms of sets:</p>
<li>Quantifiers can be expressed in set language, sort of a definition in terms of sets:
<ul>
<li>$(\forall x \in \mathbb{X}) (\phi(x)) \equiv { p \in \mathbb{X} : \phi(p) } = \mathbb{X}$</li>
<li>$(\exists x \in \mathbb{X}) (\phi(x)) \equiv { q \in \mathbb{X} : \phi(q) } \neq \emptyset$</li>
<li>$(\exists x \in \mathbb{X}) (\neg \phi(x)) \equiv \neg ( { p \in \mathbb{X} : \phi(p) } = \mathbb{X} )$</li>
<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><p>Order of quantifiers matters.</p></li>
<li>Order of quantifiers matters.</li>
</ul>
<h2 id="relations">Relations</h2>
<ul>
<li><p>Cartesian product:</p>
<li>Cartesian product:
<ul>
<li>$A \times B = { (p, q) : p \in A \land q \in B }$</li>
<li><span class="math inline">\(A \times B = \{ (p, q) : p \in A \land q \in B \}\)</span></li>
</ul></li>
<li><p>Def: A relation $R$ on a set $\mathbb X$ is a subset of $\mathbb X \times \mathbb X$ ($R \subseteq \mathbb X \times \mathbb X$)</p></li>
<li><p>Graph of a function $f()$: ${ (x, f(x) : x \in Dom(f) }$</p></li>
<li><p>Properties of:</p>
<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: $(\forall x \in \mathbb X ) (x, x) \in R \equiv (\forall x \in \mathbb X) x R x$</li>
<li>Symmetricity: $[ (\forall x, y \in \mathbb X) (x, y) \in R \implies (y, x) \in R) ] \equiv [ (\forall x, y \in \mathbb X) ( x R y \implies y R x) ]$</li>
<li>Transitivity: $(\forall x, y, z \in \mathbb X) (x R y \land y R z \implies x R z)$</li>
<li>Antisymmetricity: $(\forall x, y \in \mathbb X) (x R y \land y R x \implies x = y)$</li>
<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><p>Equivalence relations:</p>
<li>Equivalence relations:
<ul>
<li>Def: $R \subseteq \mathbb X \times \mathbb X$ is said to be an equivalence relation <em>iff</em> $R$ is reflexive, symmetric and transitive.</li>
<li>Congruence modulo n: $p R q \equiv n | p - q$</li>
<li>Def R - and equivalence relation of $\mathbb X$: The <em>equivalence class</em> of an element $x \in \mathbb X$ is the set $[x]_R = { y \in \mathbb X : x R y }$
<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 $x \in \mathbb X$ belongs to the equivalence class of some element $a$.</li>
<li>$(\forall x, y \in \mathbb X) ([x] \cap [y] \neq \emptyset \iff [x] = [y])$</li>
<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><p>Partitions</p>
<li>Partitions
<ul>
<li><p>A partition is a set containing subsets of some set $\mathbb X$ such that their collective symmetric difference equals $\mathbb X$. A partition of is a set ${ A_i: i \in \mathbb I \land A_i \subseteq \mathbb X }$ such that:</p>
<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>$(\forall x \in \mathbb X) (\exists j \in \mathbb I) (x \in A_j)$</li>
<li>$(\forall i, j \in \mathbb I) (i \neq j \implies A_i \cap A_j = \emptyset)$</li>
<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><p>${ A<em>i }</em>{i \in \mathbb I}$ is a partition <em>iff</em> there exists an equivalence relation $R$ on $\mathbb X$ such that:</p>
<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>$(\forall i \in \mathbb I) (\exists x \in \mathbb X) A_i = [x]_R$</li>
<li>$(\forall x \in \mathbb X) (\exists j \in \mathbb I) [x] = A_j$</li>
<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><p>The quotient set: $\mathbb X / R = { [a] : a \in \mathbb X }$</p></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><p>Partial orders</p>
<li>Partial orders
<ul>
<li>$\mathbb X$ is a set, $R \subseteq \mathbb X \times \mathbb X$</li>
<li><p>Def: $R$ is a partial order on $\mathbb X$ iff $R$ is:</p>
<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><p>Def: $m \in \mathbb X$ is said to be:</p>
<li>Def: <span class="math inline">\(m \in \mathbb X\)</span> is said to be:
<ul>
<li>maximal element in $(\mathbb X, \preccurlyeq)$ iff $(\forall a \in \mathbb X) m \preccurlyeq a \implies m = a$</li>
<li>largest iff $(\forall a \in \mathbb X) (a \preccurlyeq m)$</li>
<li>minimal iff $(\forall a \in \mathbb X) (a \preccurlyeq m \implies a = m)$</li>
<li>smallest iff $(\forall a \in \mathbb X) (m \preccurlyeq a)$</li>
<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><p>Def: A partial order $R$ on $\mathbb X$ is said to be <em>&ldquo;total&rdquo;</em> iff $(\forall x, y \in \mathbb X) (x R y \lor y R x)$</p></li>
<li><p>Def: A subset $B$ of $\mathbb X$ is called a chain <em>&ldquo;chain&rdquo;</em> iff $B$ is totally ordered by $R$</p>
<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>$C(\mathbb X)$ - the set of all chains in $(\mathbb X, R)$</li>
<li>A chain $D$ in $(\mathbb X, R)$ is called a maximal chain iff $D$ is a maximal element in $(C(\mathbb X), R)$</li>
<li>$K \subseteq \mathbb X$ is called an antichain in $(\mathbb X, R)$ iff $(\forall p, q \in K) (p R q \lor q R p \implies p = q)$</li>
<li>Def: $R$ is a partial order on $\mathbb X$, $R$ is called a <em>well</em> order iff $R$ is a total order on $X$ and every nonempty subset $A$ of $\mathbb X$ has the smallest element</li>
<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 $\phi$ is a propositional function defined on $\mathbb N$, if:
<li>If <span class="math inline">\(\phi\)</span> is a propositional function defined on <span class="math inline">\(\mathbb N\)</span>, if:
<ul>
<li>$\phi(1)$</li>
<li>$(\forall n \geq 1) \phi(n) \implies \phi(n+1)$</li>
<li>$(\forall k \geq 1) \phi(k)$</li>
<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>$f: \mathbb X \to \mathbb Y$</li>
<li><p>Def: $f \subseteq \mathbb X \times \mathbb Y$ is said to be a function if:</p>
<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>$(\forall x \in \mathbb X)(\exists y \in \mathbb Y) (x, y) \in f(y = f(x))$</li>
<li>$(\forall a \in \mathbb X)(\forall p, q \in \mathbb Y)((a, p) \in f \land (a, q) \in f \implies p = q)$</li>
<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><p>Types of functions $f: \mathbb X \to \mathbb Y$:</p>
<li>Types of functions <span class="math inline">\(f: \mathbb X \to \mathbb Y\)</span>:
<ul>
<li>$f$ is said to be an injection ( 1 to 1 function) iff $(\forall x_1, x_2 \in \mathbb X) x_1 \neq x_2 \implies f(x_1) \neq f(x_2)$</li>
<li>$f$ is said to be a surjection (onto function) iff $(\forall y \in \mathbb Y)(\exists x \in \mathbb X) f(x) = y$</li>
<li>If $f^{-1}$ is a function from $\mathbb Y \to \mathbb X$ then $f^{-1}$ is called the inverse function for $f$
<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: $f^{-1}$ is a function iff $f$ is a <em>bijection</em> (1 to 1 and onto)</li>
<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><p>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&rsquo;t invertible: $f^{-1}(\mathbb A)$</p></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 isnt invertible: <span class="math inline">\(f^{-1}(\mathbb A)\)</span></li>
</ul>
</div>
@ -490,17 +394,16 @@
<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>
<script>
MathJax = {
tex: {
inlineMath: [['$', '$'], ['\\(', '\\)']],
displayMath: [['$$', '$$'], ['\[', '\]']]
}
};
</script>
<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js">
</script>

View File

@ -156,9 +156,7 @@
<div class="post-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.
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>
@ -202,17 +200,16 @@
<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>
<script>
MathJax = {
tex: {
inlineMath: [['$', '$'], ['\\(', '\\)']],
displayMath: [['$$', '$$'], ['\[', '\]']]
}
};
</script>
<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js">
</script>

View File

@ -18,9 +18,7 @@
<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>
<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>
</channel>

View File

@ -164,17 +164,16 @@
<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>
<script>
MathJax = {
tex: {
inlineMath: [['$', '$'], ['\\(', '\\)']],
displayMath: [['$$', '$$'], ['\[', '\]']]
}
};
</script>
<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js">
</script>

View File

@ -156,9 +156,7 @@
<div class="post-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.
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>
@ -202,17 +200,16 @@
<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>
<script>
MathJax = {
tex: {
inlineMath: [['$', '$'], ['\\(', '\\)']],
displayMath: [['$$', '$$'], ['\[', '\]']]
}
};
</script>
<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js">
</script>

View File

@ -18,9 +18,7 @@
<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>
<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>
</channel>