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 (x,y)
that are related to (1, 2), i.e. all (x,y) such that
- (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 + 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