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

### Examples 1.4.3(c):

Let

Since two pairs **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:*(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’)]*

*(a,b)*and

*(a', b')*are related if

*a + b' = a' + b or b - a = b' - a'*

*b - a*to denote their equivalence classes. Hence:

- the symbol 2 denote the equivalence class [(1,3)] containing, for example, the pairs (1,3), (5,7), and (100, 102).
- the symbol -3 denotes the equivalence class [(4,1)], containing, for example, the pairs (4,1), (8,5), and (103, 100).

- 2 + -3 = [(1,3)] + [(103,100)] = [(1,2)] = -1

- 2 * (-3) = [(1,3)] * [(103,100)] = [(1,7)] = -6

**Z**