In this chapter we discuss equivalence relations and illustrate how they apply to basic number theory. Equivalence relations are of fundamental importance in mathematics. The epigraph by Poincare at the top of this page “says it all,” but perhaps an explanation of what he had in mind would help. As an example, consider the set of all members of a club. Group together those whose birthdays occur in the same month. Two members are thus declared “equivalent” if they belong to the same group—if their birthdays are in the same month—and the set of people in one group is called an “equivalence class.” Every club member belongs to one and only one equivalence class.
Before You Get Started
. Consider our birthday groups. What properties do you notice about our birth-month equivalence relation and about the equivalence classes? Can you think of other examples in which we group things together? Does this lead to a guess as to how you might define equivalence relations in general?
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