Soft set

{{Short description|Mathematical theory}}

Soft set theory is a generalization of fuzzy set theory, that was proposed by Molodtsov in 1999 to deal with uncertainty in a parametric manner.{{cite journal|last=Molodtsov|first=D. A.|title=Soft set theory—First results|journal=Computers & Mathematics with Applications|year=1999|volume=37|issue=4|pages=19–31|doi=10.1016/S0898-1221(99)00056-5|doi-access=free}} 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 (f, A) where A is a subset 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.{{Cite journal |last1=Alcantud |first1=José Carlos R. |last2=Khameneh |first2=Azadeh Zahedi |last3=Santos-García |first3=Gustavo |last4=Akram |first4=Muhammad |date=2024-02-25 |title=A systematic literature review of soft set theory |journal=Neural Computing and Applications |volume=36 |issue=16 |pages=8951–8975 |language=en |doi=10.1007/s00521-024-09552-x |issn=1433-3058|doi-access=free |hdl=10366/156170 |hdl-access=free }}

One of the most important steps{{Why|reason=required to justify the impact|date=April 2024}} 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.{{cite journal|last=Kharal|first=Athar|author2=B. Ahmad |title=Mappings on Soft Classes|journal=New Mathematics and Natural Computation|date=September 2011|volume=7|issue=3|pages=471–481|doi=10.1142/S1793005711002025|arxiv=1006.4940}} Soft sets have also been applied to the problem of medical diagnosis for use in medical expert systems.

Fuzzy soft sets{{Cite journal |last1=Maji |first1=P. |last2=Biswas |first2=R. |last3=Roy |first3=A. |date=2001 |title=Fuzzy soft sets |journal=J Fuzzy Math |volume=9 |pages=589–602}} and N-soft sets{{Cite journal |last1=Fatimah |first1=Fatia |last2=Rosadi |first2=Dedi |last3=Hakim |first3=R. B. Fajriya |last4=Alcantud |first4=José Carlos R. |date=2018-06-01 |title=N-soft sets and their decision making algorithms |url=https://doi.org/10.1007/s00500-017-2838-6 |journal=Soft Computing |language=en |volume=22 |issue=12 |pages=3829–3842 |doi=10.1007/s00500-017-2838-6 |issn=1433-7479}} have also been introduced. Mappings on fuzzy soft sets were defined and studied by Kharal and Ahmad.{{cite journal|last=Kharal|first=Athar|author2=B. Ahmad |title=Mappings on Fuzzy Soft Classes|journal=Advances in Fuzzy Systems|year=2009|volume=2009|pages=1–6|doi=10.1155/2009/407890|doi-access=free}}

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