fuzzy mathematics

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