Update documentation

_sources/book/mira/000-exercises.md

@@ -810,6 +810,20 @@ This completes all parts of the proof.
 ::::
 
 
+
+::::{admonition} Exercise 2.A.3
+:class: tip
+Prove that if $A, B \subseteq{R}$ and $|A| < \infty,$ then $|B \setminus A| \geq |B| - |A|.$
+
+:::{dropdown} Solution
+Suppose that $A, B \subseteq{R}$ and $|A| < \infty.$
+By the {prf:ref}`subadditivity of the outer measure<mira:thm:outer-measure-preserves-order>` and the {prf:ref}`order preserving property of the outer measure<mira:thm:outer-measure-preserves-order>`.
+
+$$|B \setminus A| + |A| \geq |A \cup B| \geq |B| \implies |B \setminus A| \geq |B| - |A|.$$
+:::
+::::
+
+
 ## Chapter 2.C
 
 ::::{admonition} Exercise 2.C.1

book/mira/000-exercises.html

@@ -1185,6 +1185,22 @@ <h2>Chapter 2.A<a class="headerlink" href="#chapter-2-a" title="Link to this hea
 This completes all parts of the proof.</p>
 </div>
 </details></div>
+<div class="tip admonition">
+<p class="admonition-title">Exercise 2.A.3</p>
+<p>Prove that if <span class="math notranslate nohighlight">\(A, B \subseteq{R}\)</span> and <span class="math notranslate nohighlight">\(|A| &lt; \infty,\)</span> then <span class="math notranslate nohighlight">\(|B \setminus A| \geq |B| - |A|.\)</span></p>
+<details class="sd-sphinx-override sd-dropdown sd-card sd-mb-3">
+<summary class="sd-summary-title sd-card-header">
+Solution<div class="sd-summary-down docutils">
+<svg version="1.1" width="1.5em" height="1.5em" class="sd-octicon sd-octicon-chevron-down" viewBox="0 0 24 24" aria-hidden="true"><path fill-rule="evenodd" d="M5.22 8.72a.75.75 0 000 1.06l6.25 6.25a.75.75 0 001.06 0l6.25-6.25a.75.75 0 00-1.06-1.06L12 14.44 6.28 8.72a.75.75 0 00-1.06 0z"></path></svg></div>
+<div class="sd-summary-up docutils">
+<svg version="1.1" width="1.5em" height="1.5em" class="sd-octicon sd-octicon-chevron-up" viewBox="0 0 24 24" aria-hidden="true"><path fill-rule="evenodd" d="M18.78 15.28a.75.75 0 000-1.06l-6.25-6.25a.75.75 0 00-1.06 0l-6.25 6.25a.75.75 0 101.06 1.06L12 9.56l5.72 5.72a.75.75 0 001.06 0z"></path></svg></div>
+</summary><div class="sd-summary-content sd-card-body docutils">
+<p class="sd-card-text">Suppose that <span class="math notranslate nohighlight">\(A, B \subseteq{R}\)</span> and <span class="math notranslate nohighlight">\(|A| &lt; \infty.\)</span>
+By the <a class="reference internal" href="002-measures.html#mira:thm:outer-measure-preserves-order">subadditivity of the outer measure</a> and the <a class="reference internal" href="002-measures.html#mira:thm:outer-measure-preserves-order">order preserving property of the outer measure</a>.</p>
+<div class="math notranslate nohighlight">
+\[|B \setminus A| + |A| \geq |A \cup B| \geq |B| \implies |B \setminus A| \geq |B| - |A|.\]</div>
+</div>
+</details></div>
 </section>
 <section id="chapter-2-c">
 <h2>Chapter 2.C<a class="headerlink" href="#chapter-2-c" title="Link to this heading">#</a></h2>