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      <text>3.1.15 (Truth of implications)</text>
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      <text><![CDATA[<p>Decide whether the following implications are true or false.</p>
<ol type="a">
  <li>If \(x\) is positive then \(\sqrt{x} \in \mathbb{R}\) (i.e. \(\sqrt{x}\) is a real number). [[input:ans1]] [[validation:ans1]] [[feedback:prt1]]</li>
  <li>If \(\sqrt{x} \in \mathbb{R}\) then \(x\) is positive. [[input:ans2]] [[validation:ans2]] [[feedback:prt2]]</li>
  <li>\(x\) is positive if and only if \(x \in \mathbb{R}\). [[input:ans3]] [[validation:ans3]] [[feedback:prt3]]</li>
</ol>]]></text>
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      <text><![CDATA[<p>The first statement is true, since the square root of any positive number is real.</p>

<p>The second statement is false. Consider the counterexample \(x = 0\). Then \(\sqrt{x} = 0 \in \mathbb{R}\) however \(0\) is not positive.</p>

<p>The third statement is also false. Recall that to see if \(A \iff B\) we must check both  the implication \(A \implies B\) and its converse \(B \implies A\) are true. In this case if we let \(A:\) \(x\) is positive and \(B:\) \(x \in \mathbb{R}\) then part a tells us that \(A \implies B\) is true, however part b tells us that \(B\) does not imply \( A\), therefore the equivalence in part c is false.</p>]]></text>
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