This paper is a Master's thesis paper in Applied Mathematics. It has two main parts. The first one is a Mathematics paper and the second one reports a scientific research in Mathematical Psychology (Cognitive Science).

The first part, from chapter 1 to chapter 6, provides a solid mathematical foundation of Fuzzy Logic Theory. Fuzzy sets are introduced and important related concepts are introduced as well, like fuzzy quantities, fuzzy relations, fuzzy operations, etc. This part is based in Nguyen (1997) , Laecio (2017) and Babuska (1998). Although the main mathematical contribution of this paper is in providing our own proofs of some important theorems in Fuzzy Theory, proofs which in the reference textbooks are shortened or only outlined. Particularly, we provide extended proofs of theorems like 2.3.4 (concerning equality of fuzzy sets), 4.1.4 (concerning isomorphism between fuzzy sets and alpha-operators), 4.1.6 (concerning the alpha-cuts and its images under some function) and 5.4.1 (fundamental equivalence between fuzzy logic and Lukasiewickz logic).

The second part, from chapter 7 to 9, provides an interesting application of Fuzzy Logic theory to Mathematical Psychology. A new model of behavior under risk called "Fuzzy Decision Model" or FDM is been developed and tested. This model is motivated and intended to be an improvement of "Prospect Theory" (PT) originally developed by Kahneman and Tversky in their famous paper "Prospect Theory: An analysis of decision under risk" (1979). The authors of this paper were awarded a Nobel prize in Economics in 2002 because of this research. Our model uses fuzzy numbers, fuzzy relations and fuzzy systems to address this difficult problem in an elegant way. Also, the model has been tested with empirical data and compared to PT, for this purpose this research is based in a previous experimental research made by the Department of Economics at University of Zurich, see Bruhin (2010). This problem provides us an interesting illustration of the potential and beauty of Fuzzy Logic theory in real world applications.