1. 1 Modeling with uncertainty requires more than probability theory There are problems where boundaries are gradual EXAMPLES: What is the boundary of the USA? Is the boundary a mathematical curve? What is a long street? What is a large real number? When do you call someone tall? Reason for using Fuzzy Sets: 1. Data reduction – driving a car, computing with language Why Fuzzy Sets?

2. 2 2. Control and fuzzy logic a. Appliances, automatic gear shifting in a car b. Subway system in Sendai, Japan (control outperformed humans in giving smoother rides) Example: Temperature control in NASA space shuttles IF x AND y THEN z is A IF x IS Y THEN z is A … etc. If the temperature is hot and increasing very fast then air conditioner fan is set to very fast and air conditioner temperature is coldest. There are four types of propositions we will study later. Why Fuzzy Sets?

3. 3 3. Fuzzy Clustering and Pattern Recognition 4. Decision making - Locate mobile telephone receptors/transmitters to optimally cover a given area - Locate recycling bins to optimally cover UCD - Position a satellite to cover the most number of mobile phone users

4. 4 Types of sets (figure from Klir&Yuan)

5. 5 VAGUENESS – lack of sharp distinction or boundaries, our ability to discriminate between different states of an event, undecidability (is a glass half full/empty) SET THEORYPROBABILITY Non-Monotonic Logics FUZZY SET DEMPSTER/SHAFER THEORY THEORY