Example: Consider the set Z of all integers. Define a relation r by saying that x and y are related if their difference y - x is divisible by m. Then we have:

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• This relation is an equivalence relation (i.e. the three conditions are satisfied)
• There are m equivalence classes:
  1. all numbers divisible by m with no remainder are in class 0
  2. all numbers divisible by m with remainder 1 are in class 1
  3. all numbers divisible by m with remainder 2 are in class 2
  4. ...
  5. all numbers divisible by m with remainder m-1 are in class m-1
• Addition can be defined by adding modulo m. That is, if we consider the equivalence classes obtained by dividing the differences by, say, 5, then we have, as an example:
  • [(2)] + [(1)] = [(3)]
  • [(2)] + [(4)] = [(1)]
  • [(3)] + [(4)] = [(3)]
  • etc...
• The (important) details are left as an exercise.

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(bgw)