Example 1.3.4(b):
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:
- This relation is an equivalence relation (i.e. the three conditions are satisfied)
- There are m equivalence classes:
- all numbers divisible by m with no remainder are in class 0
- all numbers divisible by m with remainder 1 are in class 1
- all numbers divisible by m with remainder 2 are in class 2
- ...
- 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.