Equivalence relations and mappings for crisp sets are very well known. This paper attempts an investigation of equivalence relations and mappings for fuzzy sets. We list some concepts and results related to fuzzy relations. We give some examples corresponding to the concept of fuzzy equality and fuzzy mapping introduced by Demirci [1]. In addition, we introduce the notion of preimage and quotient of fuzzy equivalence relations. Finally, we investigate relations between a fuzzy equivalence relation and a fuzzy mapping.

By using the notion of interval-valued intuitionistic fuzzy relations, we form the poset (IVIR(X), ≤ ) of intervalvalued intuitionistic fuzzy relations on a given set X. In particular, we form the subposet (IVIE(X), ≤ ) of interval-valued intuitionistic fuzzy equivalence relations on a given set X and prove that the poset (IVIE(X), ≤ ) is a complete lattice with the least element and greatest element.

Milk and dairy products are the high value foods for the elderly population. In particular, fermented milk is the best source of calcium for people in the specific age group of over 79 years. It provides a good source of protein. Regular exercise and active lifestyle are recommended to slow down the muscle loss. However, exercising without proper nutrient intake is simply not sufficient at this age. Milk and dairy products provide the iron and protein content required for effective exercise-assisted growth. Milk nutrients have the advantage of being produced in various food forms, such as liquid, semi-solid, and powder types. Fat-soluble vitamins such as retinol and vitamin K can be encapsulated using various technologies for milk and dairy products. Using the encapsulation method, spray drying and fluidized-bed coating have been used for adding the micro-nutrients to the food. Microencapsulation technology is being applied in case of the fermented dairy products too. In particular, various wall materials are being developed to enhance the viability of probiotics. In the near future, advanced high-efficiency technologies that can effectively nourish the dairy products with nutrients will be developed to produce targeted high-nutrition value food for the elderly.

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