In abstract algebra, the typical way to prove that there is an even 
number
of objects that have a certain property is to prove that they come in 
pairs. For example, Exercise 29 in Chapter 2 says to prove that the number 
of elements <i>x</i> such that <i>x<sup>2</sup></i> is not <i>e</i> is 
even.
To do this, observe that if <i>x</i> has the desired property then so does 
<i>x<sup>-1</sup></i>. But then one must also show that <i>x</i> and 
<i>x<sup>-1</sup></i> are not equal for otherwsie they do not constitute a 
pair of elements.