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Examples 1.4.3(a):

Let A be the set N x N and define an equivalence relation r on N x N and addition of the equivalence classes as follows:
  1. (a,b) is related to (a,b) if a + b = a + b
  2. [(a,b)] + [(a',b')] = [(a + a', b + b')]
  3. [(a,b)] * [(a, b)] = [(a * b + b * a, a * a + b * b)]
Which elements are contained in the equivalence classes of, say, [(1, 2)], [(0,0)] and of [(1, 0)] ? Which of the pairs (1, 5), (5, 1), (10, 14), (7, 3) are in the same equivalence classes ?
The elements in the equivalence class of [(1, 2)] are all numbers (x,y) that are related to (1, 2), i.e. all (x,y) such that In other words, the difference of the second and the first entry is one. Some members of this equivalence class are therefore Some of the elements of the class [(0,0)] and of [(1, 0)] are: Hence, (1, 1), (5, 5), (100, 100) [(0, 0)] Hence, (2, 1), (6, 5), (101, 100) [(1, 0)]. To determine which of the pairs (1, 5), (5, 1), (10, 14), (7, 3) are in the same equivalence classes, all we have to do is compare the differences between the second and the first entry: Therefore (1,5) and (10, 14) are in the same class, and (5,1) and (7,3) are also in the same class.
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