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      <text>3.1.1 (Logical Connectives)</text>
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      <text><![CDATA[<h1>Logical Connectives</h1>
<p>We have already introduced the idea of mathematical statements in the previous section and we have seen that mathematical arguments and proofs are a series of mathematical statements that follow logically from each other. However, what do we mean specifically by 'following logically'? This is what we will examine in this section. Mathematics is a language that we need to learn in order to communicate it separately. One of the barriers to writing proof is understanding and using this language. Mathematics uses a series of logical symbols as shorthand for the equivalent English statements.</p>
<p>Consider the following statements:</p>
<ul style="list-style: none;">
<li>I will go outside today.</li>
<li>I will go outside tomorrow.</li>
</ul>
<p>These are two separate statements, however we have the tools in the English language in order to combine these statements. For example we could write</p>
<ul style="list-style: none;">
<li>I will go outside today <em>and</em> I will go outside tomorrow.</li>
<li>I will go outside today <em>or</em> I will go outside tomorrow.</li>
<li>I will go outside today <em>then</em> I will go outside tomorrow.</li>
</ul>
<p>Each of these is a form of logical connective and they have important uses in mathematics. In order to link mathematical statements together we have to use a series of such <strong>logical connectives</strong>. The most common forms of logical connectives are listed in the table below.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="@@PLUGINFILE@@/Logical%20Connectives.png?time=1599220085759" alt="Table of logical connectives" width="419" height="306" /></p>
<p>We have previously used some of the symbols in our calculations but these are their formal definitions. In mathematics we illustrate the use of these connectives by the use of <em>variables</em> e.g. \(A,B,C\).</p>
<p> </p>
<h3>Example</h3>
<p>Write the statement "\((A \wedge B) \implies C"\) in English.</p>
<p>Convert the statement "If \(x\) is positive then the square root of \(x\) is real".</p>
<h3>Solution</h3>
<p>We can use the table of connectives to make these conversions. The first statement reads:</p>
<p>\(A\) and \(B\) implies \(C\).</p>
<p>We will talk more about implications in a later section.</p>
<p>To convert the second statement, we have to replace the statement with the logical symbols from the table. The statement becomes:</p>
<p>\(x&gt;0 \implies \sqrt{x} \in \mathbb{R}\).</p>
<p>Here, \(\mathbb{R}\) is the symbol for the real numbers and \(\in\) means 'is an element of'.</p>]]></text>
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