1. What are the main categories of soft set operations and their properties?
According to the research, the main categories of soft set operations are restricted soft set operations and extended soft set operations. Restricted soft set operations include basic operations such as restricted soft set operations, extended soft set operations, and complementary soft binary piecewise plus and gamma operations. Extended soft set operations include extended soft set operations, complementary extended soft set operations, and soft binary piecewise operations. The properties of these operations have been studied in detail, including their algebraic properties, distributions over other operations, and relationships with complementary operations. The research also introduced new operations and explored their properties, contributing to the literature of soft set theory.
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2. What are the two conditional complements of sets defined by Cagman, 2021?
Cagman, 2021, defined two conditional complements of sets: C-inclusive complement and C-exclusive complement. The C-inclusive complement of a set P is defined as P+C = P'C, where P' is the complement of P over the universal set U. The C-exclusive complement of a set P is defined as PC = P'C. These complements are binary operations of sets and are introduced as new complements by Sezgin et al., 2023a.
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3. How does the piecewise plus operation work in complementary soft binary?
The piecewise plus operation in complementary soft binary involves handling lP\C in the first equation. If P\C=PC', and lC' is true, then lR\C or l(CR)'. If lP\C is true, it results in lPC'R' or lPC'R. This operation is demonstrated in equation (1) and (2) of the provided section, where (K,P) * + (G,C) is equal to (K,P) ~ (G,C) [(L,R) + (K,P)]. The proof shows that K'(l), l(P\C)\R=PC'R' and K'(l) G(l), l(PC)\R=PCR, leading to the conclusion that (1)=(2).
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