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      <text>35.1.3.2 Examples intersection of sets</text>
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      <text><![CDATA[<hr>
<h4 class="HELM_example">Example</h4>
<p>Given \(A = \{3,4,5,6\}\), \(B = \{3,5,9,10,15\}\) and \(C = \{4,6,10\}\) state</p>
<ol class="HELM_parts_inline">
<li>\(A \cap B\)</li>
<li>\(B \cap C\)</li>
</ol>
and draw a Venn diagram representing these intersections.<p>
</p><h4 class="HELM_solution">Solution</h4>
<ol class="HELM_parts">
<li>The elements common to both \(A\) and \(B\) are \(3\) and \(5\).  Hence \(A \cap B = \{3,5\}\).</li>
<li>The only element common to \(B\) and \(C\) is \(10\).  Hence \(B \cap C = \{10\}\).</li>
</ol>
<img src="@@PLUGINFILE@@/35-1-fig-4-intersection_example.svg" alt="Venn diagram representing intersections" width="189" height="115" class="img-responsive atto_image_button_text-bottom"><br><p></p>
<p>The intersection of three or more sets is possible.</p>
<hr>
<h4 class="HELM_example">Example</h4>
<p>Given \(A = \{0,1,2,3\}\), \(B = \{1,2,3,4,5\}\) and \(C = \{2,3,4,7,9\}\) state </p>
<ol class="HELM_parts_inline">
<li>\(A \cap B\)</li>
<li>\((A \cap B) \cap C\)</li>
<li>\(B \cap C\)</li>
<li>\(A \cap (B \cap C)\)
</li></ol>
<h4 class="HELM_solution">Solution</h4>
<ol class="HELM_parts">
<li>The elements common to \(A\) and \(B\) are \(1, 2\) and \(3\) so \(A \cap B = \{1,2,3\}\).</li>
<li>We need to consider the sets \((A \cap B)\) and \(C\). \(A \cap B\) is given
in (a). The elements common to \((A \cap B)\) and \(C\) are \(2\) and \(3\). Hence \((A \cap B) \cap C = \{2,3\}\).</li>
<li>The elements common to \(B\) and \(C\) are \(2, 3\) and \(4\) so \(B \cap C = \{2,3,4\}\).</li>
<li>We look at the sets \(A\) and \((B \cap C)\). The common elements are \(2\) and \(3\).  Hence \(A \cap (B \cap C) = \{2,3\}\). <p></p><p>Note from (b) and (d) that \((A \cap B) \cap C = A \cap (B \cap C)\).</p></li></ol>
<p>The example illustrates a general rule. For any sets \(A\), \(B\) and \(C\) it is true that
\[(A \cap B) \cap C = A \cap (B \cap C).\]
The position of the brackets is thus unimportant. They are usually omitted and we write \(A \cap B \cap C\).</p>
<p>Suppose that sets \(A\) and \(B\) have no elements in common.  Then their intersection contains no elements and we say that \(A\) and \(B\) are <b>disjoint</b> sets. We express this as
\[A \cap B = \emptyset.\]
Recall that \(\emptyset\) is the empty set. Disjoint sets are represented by separate area regions in the Venn diagram.</p>]]></text>
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