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      <text>3.1.22 (Euler Diagrams)</text>
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      <text><![CDATA[<h3>Euler Diagrams</h3>
<p>As well as truth tables, we have other tools for understand mathematical statements and the connectives between them. One such tool is that of Euler (pronounced Oil-er) diagrams. They were invented by the prolific mathematician Leonhard Euler in the 18th century and were the inspiration for the more familiar Venn Diagrams. An <strong>Euler diagram</strong> is a diagrammatic way of understanding the hierarchy of statements and connectives. We use Euler diagrams as follows.</p>
<p> </p>
<p style="text-align: center;"><img src="@@PLUGINFILE@@/Euler%20diagrams%20use.png" alt="Diagram showing how to use Euler diagrams" width="875" height="201" /></p>
<p> </p>
<p>The images above show the representation of the statement \(A\) by an Euler diagram. The circular shape represents the statement \(A\). The rectangle enclosing the shape represents the “universe” in which the statement exists therefore when placing a dot, it must be inside the rectangle. Any dot is either inside the shape or outside it corresponding to the fact that a statement must either be true or false.</p>
<p> </p>
<h3>Example</h3>
<p>Draw the Euler diagram for the implication if \(x&lt;2\) then \(x &lt; 4\).</p>
<h3>Solution</h3>

<p>The correct diagram is</p>

<p style="text-align: center;"><img src="@@PLUGINFILE@@/Euler%20diagram%20example.png" alt="Euler diagram of the example" width="322" height="215" /></p>

<p>Note that if we think about placing a point on the diagram, we can see that if we place it in the \(x<2\) section then we must also be placing it inside the \(x < 4\) section. This helps to understand why in the truth table for implication, \(T \implies F\) is false because it is not possible to place a point on the diagram such that it is inside \(x < 2\) but not inside \(x < 4\). We can check each other row of the truth table for "\(x&lt;2\) then \(x &lt; 4\)" is possible. Therefore the Euler diagram precisely encodes the logical relationship between the two statements.</p>]]></text>
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