The document presents information on fuzzy equivalence relations. It defines fuzzy equivalence relations as fuzzy relations that are reflexive, symmetric, and transitive. Examples of fuzzy relations that satisfy these properties are provided. Related theorems state that a fuzzy relation is symmetric if its alpha cuts are symmetric relations, and transitive if its alpha cuts are transitive relations. Sample problems involving properties of compositions of fuzzy relations are outlined. The presentation concludes with a bibliography reference.

1. A PRESENTATION ON
“ FUZZY EQUIVALENCE RELATION”
PRESENTED BY-
SHRABANIKA BORUAH
ROLL NO – 20

2. CONTENTS
• OBJECTIVE
• DEFINITION
• EXAMPLES
• RELATED THEOREMS
• RELATED PROBLEMS
• BIBLIOGRAPHY

3. OBJECTIVE
• To discuss the properties of fuzzy equivalence
relation on a set
• To show a correspondence between fuzzy
equivalence relation and certain classes of
fuzzy subsets

4. DEFINITION
A crisp binary relation R(X,X) that is
reflexive,symmetric,transitive is called an
equivalence relation.
• Reflexive Property
A fuzzy relation R(X,X) is said to be reflexive ,if
µR (x,x)=1, for all xЄX
• Symmetric property
A fuzzy relation R(X,X) is said to be symmetric if
µR (x,y)= µR (y,x) for all x,yЄX

5. • Transitive property –
A fuzzy relation R(X,X) is said to be transitive if
RoR с R then,
µ RoR (x,y) ≤ µR (x,y)

6. EXAMPLES
R x1 x2 x3
x1 1 0.2 0.3
x2 0.5 1 0.2
x3 0.3 0.8 1
R x1 x2 x3
x1 0.3 1 0.5
x2 1 0.2 0.1
x3 0.5 0.1 0.9
REFLEXIVE
SYMMETRIC

7. TRANSITIVE
R x1 x2 x3
x1
x2
x3
RoR x1 x2 x3
x1
x2
x3

8. RELATED THEOREMS
• Theorem 1 : R is a symmetric fuzzy relation
iff each α-cut of R is symmetric relation
• Theorem 2 : If R is transitive fuzzy relation iff
α-cut of R is a transitive relation

9. RELATED PROBLEMS
1) If R1 and R2 are reflexive relation then R1oR2 is
a reflexive relation or not? Verify with an
example.
2) If R is symmetric then each power of R is also
symmetric.

10. BIBLIOGRAPHY
• FUZZY SETS AND FUZZY LOGIC
BY-GEORGE J.KLIR AND BO YUAN