According to our use of the word set, if C is a subset of the universal set S, then

Chapter 4, Problem 94

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According to our use of the word set, if C is a subset of the universal set S, then every element of S either does or does not belong to C. In other words, the probability of a member x of S being a member of C is either 1 (x is a member of C ) or 0 (x is not a member of C ). C is a fuzzy set if every x [ S has a probability p, 0 p 1, of being a member of C. The probability p associated with x is an estimate of the likelihood that x may belong to C when the actual composition of C is unknown. Set operations can be done on fuzzy sets as follows: If element x has probability p1 of membership in C and probability p2 of membership in D, then the probability of x being a member of C c D, C d D, and C is, respectively, max(p1, p2), min(p1, p2), and 1 p1. (If we consider the statements x [ C and x [ D as propositional wffs A and B, respectively, with certain probabilistic truth values, then the probability of x [ C c D is the probability that A ~ B is true. The rules for fuzzy set operations then parallel the rules for fuzzy logic, discussed in Exercise 54, Section 1.1.) Let S be a set of possible disease-causing agents, S = {genetics, virus, nutrition, bacteria, environment}. The fuzzy sets AIDS and ALZHEIMERS are defined as AIDS = {genetics, 0.2; virus, 0.8; nutrition, 0.1; bacteria, 0.4; environment, 0.3} and ALZHEIMERS = {genetics, 0.7; virus, 0.4; nutrition, 0.3; bacteria, 0.3; environment, 0.4}. a. Find the fuzzy set AIDS c ALZHEIMERS. b. Find the fuzzy set AIDS d ALZHEIMERS. c. Find the fuzzy set (AIDS)