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. Describe alpha-cuts corresponding to alpha = 0.4:

- for the membership functions SP (Small
Positive) which is defined as x/5 for x from 0 to 5,

2 - x/5 for x from 5 to 10, and 0 otherwise; and - for the membership functions SN (Small
Negative) which is defined as -x/5 for x from -5 to 0,

2 + x/5 for x from -10 to -5, and 0 otherwise.

2. For x_{1} = Small Positive and
x_{2} = Small Negative (as in Problem 1), find the alpha-cut for y =
x_{1} - x_{2} for alpha = 0.4.

3-4. Find the range of the function
y = (x_{1} + 1)^{2} - x_{1} *
x_{2} when x_{1} is between 0 and 1, and
x_{2} is between -2 and 1, by
using two methods:

- the calculus-based method for finding the exact range, and
- the naive interval computations method for computing the enclosure for the range.

5. Use fuzzy optimization to find the price of the cheapest healthy lunch at the Union. The objective function is the price, which goes from $2 (sandwich) to $10 (chicken salad); as usual, use linear interpolation to find the membership function corresponding to "minimal". To get a membership function for "healthy", use linear interpolation; assume that the $2 lunch is absolutely un-healthy, while the $10 lunch is absolutely healthy.

6. Use the crisp clustering algorithm to cluster the following 1-D data: objects are characterized by values 0.0, 1.0, 5.0, 6.0, and 7.0, we have two clusters, and the initial representatives are 0.0 and 7.5.

7. On a line x_{1} + x_{2} = 1, find a
point which is the closest to the point (3,4). Use the Lagrange
multipliers method.

8. Use the fuzzy clustering algorithm to cluster the data from Problem 6. Use m = 2, and run only one iteration; it if Ok not to perform computations, but you need to write down all the formulas for each step.

9-10. Given the following number of people in each age bracket: [20, 30]: 2, [30, 40]: 1, [40, 50]: 2, compute intervals for the mean age and the variance. Lower bound for the variance is for extra credit.