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<h1 class="title">Naive Set Theory</h1>
<address class="author">Mort Yao</address>
<!-- h3 class="date">2017-04-02</h3 -->
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<p>Basic set theory, with ZF axioms:</p>
<ul>
<li>Paul Halmos. <strong><em>Naive Set Theory.</em></strong></li>
</ul>
<hr />
<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 (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 <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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