Soft set theory is a generalization of
fuzzy set theory
In mathematics, fuzzy sets (a.k.a. uncertain sets) are sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh in 1965 as an extension of the classical notion of set.
At the same time, defined ...
, that was proposed by Molodtsov in 1999 to deal with uncertainty in a parametric manner. A soft set is a parameterised family of sets - intuitively, this is "soft" because the boundary of the set depends on the parameters. Formally, a soft set, over a universal set X and set of parameters E is a
pair
Pair or PAIR or Pairing may refer to:
Government and politics
* Pair (parliamentary convention), matching of members unable to attend, so as not to change the voting margin
* ''Pair'', a member of the Prussian House of Lords
* ''Pair'', the Frenc ...
(''f'', ''A'') where ''A'' is a
subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
of E, and ''f'' is a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
from ''A'' to the power set of X. For each ''e'' in ''A'', the set ''f''(''e'') is called the value set of ''e'' in (''f'', ''A'').
One of the most important steps for the new theory of soft sets was to define mappings on soft sets, which was achieved in 2009 by the mathematicians Athar Kharal and Bashir Ahmad, with the results published in 2011. Soft sets have also been applied to the problem of medical diagnosis for use in medical expert systems.
Fuzzy soft sets have also been introduced. Mappings on fuzzy soft sets were defined and studied by Kharal and Ahmad.
Notes
References
*Molodtsov D. A.'' A theory of soft sets''. Moscow: Editorial URSS, 2004.
*Matsievsky S. V.'' Sets, multisets, fuzzy and soft sets without universe''. Vestnik IKSUR, 2007, N. 10, pp. 44–52.
*Ahmad, B., Kharal, A.
On Fuzzy Soft Sets'. Advances in Fuzzy Systems Volume 2009 (2009), Article ID 586507, 6 pages .
Set theory
{{Mathlogic-stub