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The human cognition and interaction with the outside world involves structure with no sharp boundaries in which the transition of membership to non-membership function is gradual rather than abrupt. The concept of cut sets and fuzzy numbers were developed and intensive research has been taking place and applied in human cognition. In this paper, we have introduced matrix-cut for trapezoidal fuzzy number by studying its properties, arithmetic operations and decomposition theorems. Finally, the Trapezoidal Intuitionistic Fuzzy number (TFN) and its arithmetic properties using matrix cut have also been proposed.

The Interval-valued intuitionistic fuzzy sets (IVIFSs) have long been used to model vagueness and have been used in decision making, pattern recognition, image processing, and other applications. In this study, IVIFSs are defined using Gaussian membership functions (GMFs), and new measures of the distance, the overlap, and the angle between two sets are developed. The proposed methodology is used to determine the similarities between test subjects in genetic brain profiling.

Fuzzy sets have been extensively researched and results have been developed based on the extensions of fuzzy sets. In this chapter, fuzzy sets and its extensions are discussed. Z-numbers along with weighted sum product assessment method is used to obtain a feasible solution to the location selection problem for installation of smog towers in a densely populated locality. The degrees of freedom namely degree of membership, degree of non-membership and the degree of hesitancy have been expressed as Zadeh’s Z-number with probability quotient for the degrees. Further, ranking of the alternatives based on Z-numbers and WASPAS to allocate smog towers to residential areas stricken by air pollution.

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