then ‘∼’ is just a common equivalence relation on X. The corresponding quotient space is denoted by [X].
(2.10)
That is, d satisfies the triangle inequality and is a distance function. is a metric space.
is a common equivalence relation on X, and is called a sectional relationship of .
is an equivalence relation with as its base. Let . Then define a fuzzy equivalence relation on X uniquely, and with as its cut relation.
does not satisfy the condition (3) in the definition of fuzzy equivalence relation.
rii | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 |
1 | 1 | |||||||||||||
2 | 0 | 1 | ||||||||||||
3 | 0 | 0 | 1 | |||||||||||
4 | 0 | 0 | 0.4 | 1 | ||||||||||
5 | 0 | 0.8 | 0 | 0 | 1 | |||||||||
6 | 0.5 | 0 | 0.2 | 0.2 | 0 | 1 | ||||||||
7 | 0 | 0.8 | 0 | 0 | 0.4 | 0 | 1 | |||||||
8 | 0.4 | 0.2 | 0.2 | 0.4 | 0 | 0.8 | 0 | 1 | ||||||
9 | 0 | 0.4 | 0 | 0.8 | 0.4 | 0.2 | 0.4 | 0 | 1 | |||||
10 | 0 | 0 | 0.2 | 0.2 | 0 | 0 | 0.2 | 0 | 0.2 | 1 | ||||
11 | 0 | 0.5 | 0.2 | 0.2 | 0 | 0 | 0.8 | 0 | 0.4 | 0.2 | 1 | |||
12 | 0 | 0 | 0.2 | 0.8 | 0 | 0 | 0 | 0 | 0.4 | 0.8 | 0 | 1 | ||
13 | 0.8 | 0 | 0.2 | 0.4 | 0 | 0.4 | 0 | 0.4 | 0 | 0 | 0 | 0 | 1 | |
14 | 0 | 0.8 | 0 | 0.2 | 0.4 | 0 | 0.8 | 0 | 0.2 | 0.2 | 0.6 | 0 | 0 | 1 |