## 1.4. Natural Numbers, Integers, and Rational Numbers

### 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:

If you add [(1,2)] + [(4, 6)] you would get the following:
*(a,b)*is related to*(a’,b’)*if*a + b’ = a’ + b**[(a,b)] + [(a',b')] = [(a + a', b + b')]**[(a,b)] * [(a’, b’)] = [(a * b’ + b * a’, a * a’ + b * b’)]*

- [(1,2)] + [(4, 6)] = [(1 + 4, 2 + 6)] = [(5, 8)]

- [(1,2)] contains all pairs whose difference y - x = 1
- [(4,6)] contains all pairs whose difference y - x = 2
- [(1,2)] + [(4,6)] contains all pairs whose difference y - x = 3

- [(3,1)] + [(1,3)] = [(3+1, 1+3)]

- [(3,1)] contains all pairs whose difference y - x = -2
- [(1,3)] contains all pairs whose difference y - x = 2
- [(3,1)] + [(1,3)] contains all pairs whose difference y - x = 0.

- [(5,4)] * [(7, 4)] = [(5*4 + 4*7, 5*7 + 4*4)] = [(48,51)]

- [(5,4)] contains all pairs whose difference y - x is -1
- [(7,4)] contains all pairs whose difference y - x is -3
- [(5,4)] * [(7, 4)] = [(48,51)] contains all pairs whose difference y - x = 3

- [(1,2)] * [(2,1)] = [(1*1 + 2*2, 1*2 + 2*1)] = [(5,4)]

- [(1,2)] contains all pairs whose difference y - x is 1
- [(2,1)] contains all pairs whose difference y - x is -1
- [(1,2)] * [(2,1)] = [(5,4)] contains all pairs whose difference y - x is -1.