This paper proposes a novel framework for bounded intuitionistic fuzzy sets based on the use of lower and upper threshold parameters. Unlike classical cut-set approaches, which produce crisp subsets and lead to loss of information, the proposed method preserves the original membership and nonmembership degrees while restricting the universe to elements satisfying given threshold conditions. The main objective of this study is to develop a value-preserving structure that allows a more refined analysis of intuitionistic fuzzy sets. Within this framework, several types of bounded intuitionistic fuzzy sets are introduced and their fundamental algebraic properties are systematically investigated. In particular, inclusion relations, set-theoretic operations, and behavior under mappings are examined in detail. Furthermore, the proposed approach is applied to intuitionistic fuzzy subgroup theory. It is shown that bounded intuitionistic fuzzy sets derived from intuitionistic fuzzy subgroups retain subgroup properties within the corresponding restricted structures. Additional characterization results are also obtained. The proposed framework generalizes existing approaches and provides a flexible and effective tool for the analysis of intuitionistic fuzzy structures. It offers a more precise way to handle uncertainty while maintaining the inherent graded information, and it may be useful in various applications such as algebraic fuzzy systems, decision-making, and information processing.
Ümit Deniz (Fri,) studied this question.