1. Fuzzy Relation Equation

2. Problem Definition
Notations
Problem

3. Meaning of Relation Equation
Suppose we know the overall relation between X and Z, and a part of the relation between Y and Z, what should be between X and Y.

4. Problem Partition
Partitioned solution set

5. Solution
Condition
Example

6. Solution Method
Solution Lattice
Maximum Solution

7. Solution Method
Maximum Solution

8. Solution Method
Minimal Solution
Reduced Problems
Solution Method

9. Solution Method

10. SUP-I Fuzzy Relation Equation
Solution Set
Example

11. INF-Omega I Fuzzy Relation Equation
Solution Set
Example
Other Type of Problem

12. Some Measures of Set Equality
Equality Index
The equality index can be used as a goodness measure of approximate solution of relation equations.

13. Application
Example:
A system can be represented as a fuzzy relation, which can be decomposed into two subsystem, the one is reachable, but the other is unreachable. Suppose the overall system and the reachable subsystem can be known and modeled by fuzzy relations. Find the relation of the other subsystem.