Name: _____________________________________________________________

5 pages of notes allowed.

Please place the solution of each problem on a separate sheet of paper, with your name on top of each sheet.

1. Suppose that 7 people out of 10 think that the new Computer Science building is cool. What degree of certainty would you assign to the statement that this building is cool?

2. Suppose that, as a reply to the question "How interesting is the class?" a student marked 4 on a 0 to 5 scale. What degree of certainty would you assign to the statement that this class is interesting?

3. Some classes are medium-size, i.e., they have a reasonable number of student but not too many. Assume that everyone agrees that 10 is not medium size, 15 absolutely is medium size, and 25 is absolutely not medium size. Use linear interpolation to describe the membership function for the property "medium size".

4. Suppose that a student's degree of confidence that the new CS building is cool is 0.8, and that her belief that this building is comfortable is 0.7. Estimate the student's degree of confidence that this building is cool AND comfortable by using min and algebraic product t-norms.

5. Is the sum (a + b) a fuzzy "or"-operation? If your answer
is "yes", show that it satisfies all the requirements of an
"or"-operation. If your answer is "no", explain which requirements
are satisfied and which are not satisfied. *For extra credit:*
how to modify the sum so that it becomes a fuzzy "or"-operation?

6-7. For a professor teaching a class, to describe how big should be the letters that this professor writes on the board, we have two rules:

- if the class is small, it is OK to have small-size letters;
- if the class is large, the professor should write big-size letters.

- For "class is small": the value is 1 - x / 25 when x < 25 and 0 otherwise.
- For "class is large": the value is x / 25 when x < 25 and 1 otherwise.
- For "letter is small": the value is 1 - s / 4 when s < 4 and 0 otherwise.
- For "letter is big": the value is s / 4 when s < 4 and 1 otherwise.

8. Use the Least Squares method to estimate the average speed v of the car based on the following two measurements:

- after 1 hour (t
_{1}= 1), the car traveled for approximately d_{1}= 35 miles; - after 2 hours (t
_{2}= 2), the car traveled for approximately d_{2}= 65 miles.

9. Write down a formula for the centroid defuzzification. Explain
why this formula is not always applicable. *For extra credit:*
explain the derivation of the centroid defuzzification formula.