math/set: update

Author: soimort

math/set/index.html

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-    <meta name="dcterms.date" content="2016-12-15">
+    <meta name="dcterms.date" content="2017-04-02">
     <title>Naive Set Theory</title>
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       <header>
         <h1 class="title">Naive Set Theory</h1>
         <address class="author">Mort Yao</address>
-        <!-- h3 class="date">2016-12-15</h3 -->
+        <!-- h3 class="date">2017-04-02</h3 -->
       </header>
       <div id="content">
 <p>Basic set theory, with ZF axioms:</p>
 <ul>
 <li>Paul Halmos. <strong><em>Naive Set Theory.</em></strong></li>
 </ul>
 <hr />
-<p><strong>Countable and uncountable sets.</strong> A set is called <em>countable</em> if its elements can be enumerated; otherwise, it is called <em>uncountable</em>.</p>
-<p>Clearly, all finite sets are countable. The set of integers and the set of rational numbers are also countable. However, the set of real numbers <span class="math inline">\(\mathbb{R}\)</span> is uncountable.</p>
+<p><strong>Finite sets and infinite sets</strong>. A set is called <em>finite</em> if it contains finitely many elements; otherwise, it is called <em>infinite</em>.</p>
+<p><strong>Countable set and uncountable sets.</strong> A set is called <em>countable</em> if its elements can be enumerated; otherwise, it is called <em>uncountable</em>.</p>
+<p>Clearly, all finite sets are countable. The set of natural numbers <span class="math inline">\(\mathbb{N}\)</span>, the set of integers <span class="math inline">\(\mathbb{Z}\)</span> and the set of rational numbers <span class="math inline">\(\mathbb{Q}\)</span> are also countable. However, the set of real numbers <span class="math inline">\(\mathbb{R}\)</span> is uncountable.</p>
 <p><strong>Subset and superset.</strong> <span class="math inline">\(A\)</span> is a <em>subset</em> of <span class="math inline">\(B\)</span> (or: <span class="math inline">\(B\)</span> is a <em>superset</em> of <span class="math inline">\(A\)</span>), denoted as <span class="math inline">\(A \subseteq B\)</span> (or: <span class="math inline">\(B \supseteq A\)</span>), if and only if for every <span class="math inline">\(x \in A\)</span>, there is <span class="math inline">\(x \in B\)</span>.</p>
 <p><span class="math inline">\(A\)</span> and <span class="math inline">\(B\)</span> are said to be equal, denoted as <span class="math inline">\(A = B\)</span>, if and only if <span class="math inline">\(A \subseteq B\)</span> and <span class="math inline">\(B \subseteq A\)</span>; otherwise, <span class="math inline">\(A\)</span> and <span class="math inline">\(B\)</span> are said to be unequal, denoted as <span class="math inline">\(A \neq B\)</span>.</p>
 <p><span class="math inline">\(A\)</span> is a <em>proper subset</em> of <span class="math inline">\(B\)</span> (or: <span class="math inline">\(B\)</span> is a <em>proper superset</em> of <span class="math inline">\(A\)</span>), denoted as <span class="math inline">\(A \subset B\)</span> (or: <span class="math inline">\(B \supset A\)</span>), if and only if <span class="math inline">\(A \subseteq B\)</span> and <span class="math inline">\(A \neq B\)</span>.</p>
 <p><strong>Union.</strong> <span class="math inline">\(A \cup B = \{ x : x \in A \lor x \in B \}\)</span>.</p>
 <p><strong>Intersection.</strong> <span class="math inline">\(A \cap B = \{ x : x \in A \land x \in B \}\)</span>.</p>
 <p><strong>Difference.</strong> <span class="math inline">\(A \setminus B = \{ x : x \not\in A \land x \in B \}\)</span>.</p>
 <p><strong>Symmetric difference.</strong> <span class="math inline">\(A \triangle B = (A \setminus B) \cup (B \setminus A) = \{ x : x \in A \oplus x \in B \}\)</span>.</p>
-<p><strong>Cartesian product.</strong> <span class="math inline">\(A \times B = \{ (x,y) : x \in A \land y \in B \}\)</span>.</p>
+<p><strong>Cartesian product (cross product).</strong> <span class="math inline">\(A \times B = \{ (x,y) : x \in A \land y \in B \}\)</span>.</p>
 <p><strong>Power set.</strong> <span class="math inline">\(\mathcal{P}(A) = \{ X : X \subseteq A \}\)</span>.</p>
-<p><strong>Empty set.</strong> The empty set is denoted as <span class="math inline">\(\varnothing\)</span>. <span class="math inline">\(| \varnothing | = 0\)</span>.</p>
+<p><strong>Empty set.</strong> The empty set <span class="math inline">\(\{\}\)</span> is denoted as <span class="math inline">\(\varnothing\)</span>. <span class="math inline">\(| \varnothing | = 0\)</span>.</p>
 <p><strong>Disjoint sets.</strong> Two sets <span class="math inline">\(A\)</span> and <span class="math inline">\(B\)</span> are said to be <em>disjoint</em>, if and only if <span class="math inline">\(A \cap B = \varnothing\)</span>.</p>
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math/set/src.md

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 % Naive Set Theory
 % Mort Yao
-% 2016-12-15
+% 2017-04-02
 
 Basic set theory, with ZF axioms:
 
 * Paul Halmos. ***Naive Set Theory.***
 
 ---
 
-**Countable and uncountable sets.** A set is called *countable* if its elements can be enumerated; otherwise, it is called *uncountable*.
+**Finite sets and infinite sets**. A set is called *finite* if it contains finitely many elements; otherwise, it is called *infinite*.
 
-Clearly, all finite sets are countable. The set of integers and the set of rational numbers are also countable. However, the set of real numbers $\mathbb{R}$ is uncountable.
+**Countable set and uncountable sets.** A set is called *countable* if its elements can be enumerated; otherwise, it is called *uncountable*.
+
+Clearly, all finite sets are countable. The set of natural numbers $\mathbb{N}$, the set of integers $\mathbb{Z}$ and the set of rational numbers $\mathbb{Q}$ are also countable. However, the set of real numbers $\mathbb{R}$ is uncountable.
 
 **Subset and superset.** $A$ is a *subset* of $B$ (or: $B$ is a *superset* of $A$), denoted as $A \subseteq B$ (or: $B \supseteq A$), if and only if for every $x \in A$, there is $x \in B$.
 
@@ -26,10 +28,10 @@ $A$ is a *proper subset* of $B$ (or: $B$ is a *proper superset* of $A$), denoted
 
 **Symmetric difference.** $A \triangle B = (A \setminus B) \cup (B \setminus A) = \{ x : x \in A \oplus x \in B \}$.
 
-**Cartesian product.** $A \times B = \{ (x,y) : x \in A \land y \in B \}$.
+**Cartesian product (cross product).** $A \times B = \{ (x,y) : x \in A \land y \in B \}$.
 
 **Power set.** $\mathcal{P}(A) = \{ X : X \subseteq A \}$.
 
-**Empty set.** The empty set is denoted as $\varnothing$. $| \varnothing | = 0$.
+**Empty set.** The empty set $\{\}$ is denoted as $\varnothing$. $| \varnothing | = 0$.
 
 **Disjoint sets.** Two sets $A$ and $B$ are said to be *disjoint*, if and only if $A \cap B = \varnothing$.