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

### 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:

The elements in the equivalence class of [(1, 2)] are all numbers *(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’)]*

*(x,y)*that are related to (1, 2), i.e. all

*(x,y)*such that

*1 + y = x + 2*or*y - x = 1*

- (2, 3), (3, 4), (100, 101) [(1, 2)]

*(x, y)*[(0, 0)] if (0, 0) ~*(x,y)**0 + y = x + 0*or*y = x*

*(x, y)*(1, 0) if (1, 0) ~*(x, y)**1 + y = 0 + x*or*x - y = 1*

- (1,5): the difference
*y - x*= 4 - (5, 1): the difference
*y - x*= -4 - (10, 14): the difference
*y - x*= 4 - (7, 3): the difference
*y - x*= -4