Soft set theory is a generalization of
fuzzy set theory
Fuzzy or Fuzzies may refer to:
Music
* Fuzzy (band), a 1990s Boston indie pop band
* Fuzzy (composer), Danish composer Jens Vilhelm Pedersen (born 1939)
* ''Fuzzy'' (album), 1993 debut album of American rock band Grant Lee Buffalo
* "Fuzzy", a ...
, 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 Fren ...
(''f'', ''A'') where ''A'' is a
subset
In mathematics, a 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 a ...
of E, and ''f'' is a
function 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'').
A systematic literature review on soft set theory was published in the journal Neural Computing and Applications in February 2024.
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 and N-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