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      <text>3.2.12 (Counter-examples)</text>
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      <text><![CDATA[<h1>Examples and counter examples</h1>

<p>The scientific method is the cornerstone of all scientific research. It relies on substantive experimental evidence. However, in mathematics, we do not consider such evidence as constituting a full proof. The reason for this is that the production of one single <b>counter example</b> which shows a statement to be false, it is sufficient to <i>disprove</i> the statement. This is because <i>disproving</i> a statement is logically equivalent to proving its <i>negation</i>. For example the famous Riemann hypothesis has been shown to hold for approximately the first \(10,000,000,000,000\) solutions but is not considered proven. Incidentally if you prove the Riemann hypothesis, you will win $1 million from the Clay mathematics institute.</p>

<h3>Example</h3>

<p>For any numbers \(a\) and \(b\), if \(a^2 = b^2\) then \(a=b\).</p>

<h3>Solution</h3>

<p>Note that this is a statement with a universal quantifier 'for any'. Therefore if we find a single example of numbers \(a\) and \(b\) for which the statement is false. So we take \(a = 2\) and \(b = -2\). Then \(a^2 = 4\) and \(b^2 = 4\), however, this doesn't mean that \(a = b\) because we already know that \(a \not= b\).</p>

<p>Is this the only counter example that would have worked?</p>
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<p>As well as the use of counter examples to disprove certain statements, the idea of an example is important to that of proof. It an important skill to be able to <b>exemplify</b> the result of a given theorem. By this we mean that it is often useful to provide a specific example of a result. This can be helpful for getting an idea to formulating a proof or it can be the result of an <i>existence</i> proof. For example if we wish to prove that the equation \(x+1\) is always an even number whenever \(x\) is odd, then it may be helpful to note the first few examples that hold are \(1+1 = 2, 3+1 = 3, \dots\). Alternatively some proofs revolve around trying to prove a specific example of something exists. For example consider the statement 'there exists a prime number \(p\)' such that \(p + 8\) is also prime. If we set \(p =3\) we see that this satisfies the condition, so it is an <i>example</i> which proves the statement.</p>]]></text>
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