# Upper and lower set formulas: restriction and modification of the Dempster-Pawlak formalism

International Journal of Applied Mathematics and Computer Science (2002)

- Volume: 12, Issue: 3, page 359-369
- ISSN: 1641-876X

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topTürkşen, Ismail. "Upper and lower set formulas: restriction and modification of the Dempster-Pawlak formalism." International Journal of Applied Mathematics and Computer Science 12.3 (2002): 359-369. <http://eudml.org/doc/207593>.

@article{Türkşen2002,

abstract = {A modification of Dempster's and Pawlak's constructs forms a new foundation for the identification of upper and lower sets formulas. Also, in this modified Dempster-Pawlak construct we require that subsets of the power set be restricted to the well-known information granules of the power set. An aggregation of upper information granules amongst each other and lower information granules amongst each other determine upper and lower set formulas for both crisp and fuzzy sets. The results are equivalent to the Truth Table derivation of FDCF and FCCF, Fuzzy Disjunctive Canonical Forms and Fuzzy Conjunctive Canonical Forms, respectively. Furthermore, they collapse to , i.e., the equivalence of Disjunctive Normal Forms and Conjunctive Normal Forms, in the combination of concepts once the LEM, LC and absorption, idempotency and distributivity axioms are admitted into the framework. Finally, a proof of the containment is obtained between FDCF and FCCF for the particular class of strict and nilpotent Archimedian -norms and -conorms.},

author = {Türkşen, Ismail},

journal = {International Journal of Applied Mathematics and Computer Science},

keywords = {fuzzy canonical formulas; Dempster-Pawlak modification; upper and lower set formulas; information granules},

language = {eng},

number = {3},

pages = {359-369},

title = {Upper and lower set formulas: restriction and modification of the Dempster-Pawlak formalism},

url = {http://eudml.org/doc/207593},

volume = {12},

year = {2002},

}

TY - JOUR

AU - Türkşen, Ismail

TI - Upper and lower set formulas: restriction and modification of the Dempster-Pawlak formalism

JO - International Journal of Applied Mathematics and Computer Science

PY - 2002

VL - 12

IS - 3

SP - 359

EP - 369

AB - A modification of Dempster's and Pawlak's constructs forms a new foundation for the identification of upper and lower sets formulas. Also, in this modified Dempster-Pawlak construct we require that subsets of the power set be restricted to the well-known information granules of the power set. An aggregation of upper information granules amongst each other and lower information granules amongst each other determine upper and lower set formulas for both crisp and fuzzy sets. The results are equivalent to the Truth Table derivation of FDCF and FCCF, Fuzzy Disjunctive Canonical Forms and Fuzzy Conjunctive Canonical Forms, respectively. Furthermore, they collapse to , i.e., the equivalence of Disjunctive Normal Forms and Conjunctive Normal Forms, in the combination of concepts once the LEM, LC and absorption, idempotency and distributivity axioms are admitted into the framework. Finally, a proof of the containment is obtained between FDCF and FCCF for the particular class of strict and nilpotent Archimedian -norms and -conorms.

LA - eng

KW - fuzzy canonical formulas; Dempster-Pawlak modification; upper and lower set formulas; information granules

UR - http://eudml.org/doc/207593

ER -

## References

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- Pawlak Z. (1991): Rough Sets. - Dordrecht: Kluwer. Zbl0758.68054
- Resconi G., Türkşen I.B. (2001): Canonical forms of fuzzy truthoods by meta-theory based upon modal logic. - Inf. Sci., Vol. 131, pp. 157-194. Zbl1004.03021
- Türkşen I.B. (1986): Interval-valued fuzzy sets based on normal forms. - Fuzzy Sets Syst., Vol. 20, pp. 191-210. Zbl0618.94020
- Türkşen I.B. (1992): Interval-valued fuzzy sets and compensatory AND. - Fuzzy Sets Syst., Vol. 51, pp. 87-100.
- Türkşen I.B. (1999): Theories of set and logic with crisp or fuzzy information granules. - J. Adv. Comp. Intell., Vol. 3, No. 4, pp. 264-273.
- Türkşen I.B. (2001): Computing with descriptive and veristic words: Knowledge representation and reasoning, In: Computing With Words (P.P.Wang, Ed.). - New York: Wiley, pp. 297-328.

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