Operations on soft sets revisited

Soft sets, as a mathematical tool for dealing with uncertainty, have recently gained considerable attention, including some successful applications in information processing, decision, demand analysis, and forecasting. To construct new soft sets from given soft sets, some operations on soft sets have been proposed. Unfortunately, such operations cannot keep all classical set-theoretic laws true for soft sets. In this paper, we redefine the intersection, complement, and difference of soft sets and investigate the algebraic properties of these operations along with a known union operation. We find that the new operation system on soft sets inherits all basic properties of operations on classical sets, which justifies our definitions.

💡 Research Summary

The paper addresses a long‑standing inconsistency in soft‑set theory: the classical set‑theoretic laws (commutativity, associativity, distributivity, De Morgan’s laws, etc.) are not fully satisfied by the previously proposed operations on soft sets. Soft sets are defined as pairs ((E,F)) where (E) is a set of parameters and (F:E\rightarrow\mathcal P(U)) maps each parameter to a subset of a universal object set (U). Existing definitions of intersection, complement and difference restrict the resulting parameter set to the intersection of the original parameter sets, causing loss of information when the parameter sets are disjoint and leading to violations of basic algebraic identities.

To remedy this, the authors propose a global‑parameter approach. For any two soft sets (\mathcal F_1=(E_1,F_1)) and (\mathcal F_2=(E_2,F_2)) they first fix a common parameter universe (E=E_1\cup E_2) (or a larger ambient parameter space). The new operations are defined on this unified parameter set:

• Union remains the familiar pointwise union: ((F_1\cup F_2)(e)=F_1(e)\cup F_2(e)) for every (e\in E), interpreting undefined values as the empty set.
• Intersection is defined pointwise but retains the values of each operand on parameters that are not shared:
  – If (e\in E_1\cap E_2), ((F_1\cap F_2)(e)=F_1(e)\cap F_2(e)).
  – If (e\in E_1\setminus E_2), ((F_1\cap F_2)(e)=F_1(e)).
  – If (e\in E_2\setminus E_1), ((F_1\cap F_2)(e)=F_2(e)).
• Complement is taken with respect to the whole universe (U) and the whole parameter set (E): (\mathcal F^{c}=(E,,U\setminus F(e))). This mirrors the classical complement operation.
• Difference is then defined as (\mathcal F_1\setminus\mathcal F_2=\mathcal F_1\cap\mathcal F_2^{c}).

With these definitions the authors systematically prove that the soft‑set algebra inherits all the elementary properties of classical set algebra:

1. Commutativity of both union and intersection.
2. Associativity for both operations.
3. Distributivity of intersection over union and vice‑versa.
4. De Morgan’s laws: ((\mathcal F_1\cup\mathcal F_2)^{c}=\mathcal F_1^{c}\cap\mathcal F_2^{c}) and ((\mathcal F_1\cap\mathcal F_2)^{c}=\mathcal F_1^{c}\cup\mathcal F_2^{c}).
5. Identity and null elements: the empty soft set and the universal soft set act as neutral elements for union and intersection respectively.
6. Boolean algebra structure: the collection of all soft sets over a fixed parameter universe (E) and object universe (U) forms a Boolean algebra isomorphic to the power set algebra (\mathcal P(E\times U)).

These results demonstrate that the newly introduced operation system resolves the earlier logical gaps and provides a mathematically sound foundation for soft‑set manipulation. The paper also discusses practical implications. Because the parameter set is no longer trimmed during operations, dynamic environments—where parameters may be added or removed during decision‑making, data‑mining, or risk‑assessment processes—retain full information throughout. Consequently, algorithms that rely on repeated set operations (e.g., multi‑criteria decision analysis, soft‑set based clustering) can be designed without worrying about inadvertent loss of parameter data.

In the concluding section the authors outline future research directions: extending the framework to fuzzy‑soft sets, exploring efficient computational representations of the global‑parameter soft sets, and conducting empirical studies on real‑world datasets to validate the theoretical advantages. Overall, the paper makes a substantial contribution by aligning soft‑set operations with classical set theory, thereby enhancing both the theoretical elegance and the applicability of soft‑set methods in handling uncertainty.