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Examples 1.4.3(b):

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)]
Below are some examples for addition and multiplication of these equivalence classes.
If you add [(1,2)] + [(4, 6)] you would get the following: By the above examples, that implies Adding [(3,1)] + [(1,3)] gives the following: This is, by the above example, equivalent to the following: Multiplying the equivalence classes [(5,4)] * [(7, 4)] we get the following: This is, by the above example, equivalent to the following: Multiplying the equivalence classes [(1,2)] * [(2,1)] we get the following: This is, by the above example, equivalent to the following:
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