Reasoning with Imprecision

Fuzzy Sets [Lug Ch.8.2.2]
- The items in a fuzzy set have a membership function to represent the degree of membership of each item.
- Example:
  - tall = {0 if height<5, (height-5)/2 if 5<=height<=7, 1 if height>7}
  - medium = {0 if height<3, height-3 if 3<=height<=4, 1 if 4<=height<5, 6-height if 5<=height<=6, 0 if height>6}
  - short = {1 if height<2, (4-height)/2 if 2<=height<=4, 0 if height>4}
  - A graph of this looks like:
```
     Short    Medium    Tall
1.0 +----+     +--+     +-------------------
    |     \    |  |    /
    |      \  /    \  /
0.5 +       \|      |/
    |       /\      /\
    |       | \    / |
0.0 +-+--+--+--+--+--+--+-------------------
      |     |     |     |
     1.0   3.0   5.0   7.0
               
```
  - Here are some example values:
```
Height    shortness mediumness tallness
--------------------------------------
1' 2"     1.00      0.00       0.00
2' 6"     0.75      0.00       0.00
3' 2"     0.42      0.16       0.00
4' 5"     0.00      1.00       0.00
5' 9"     0.00      0.25       0.38
6' 1"     0.00      0.00       0.54
7' 2"     0.00      0.00       1.00
               
```
- Expressions like "A is X" can be interpreted as degrees of truth, e.g., "5'9" is tall" = 0.38.
- Membership functions used in most applications almost never have such simple shapes
- Also (occasionally), may have multiple criteria: tall depends on both a person's height and their age (he's tall for his age). A two-dimensional membership function, or a "fuzzy relation"
- Logic operations
  - truth(~X) = 1.0 - truth(X)
  - truth(X & Y) = minimum(truth(X), truth(Y))
  - truth(X | Y) = maximum(truth(X), truth(Y))
- Examples
  - Definition of tall as above
  - old = {0 if age<18, (age-18)/42 if 18<=age<=60, 1 if age > 60}
  - Here are some example values:
```
Height Age  tall  old  tall & old tall | old ~tall
------------------------------------------------
1' 2"   12  0.00  0.00    0.00       0.00    1.00
3' 2"   50  0.00  0.76    0.00       0.76    1.00
5' 9"   65  0.38  1.00    0.38       1.00    0.62
7' 2"   30  1.00  0.28    0.28       1.00    0.00
               
```

Exam Style Questions

Given the following fuzzy set membership functions, and a fruit that is 225cm above ground, sells for J$300 per kilogram, and has a low sugar content, is:

harvestable and cheap and sweet
harvestable or cheap or sweet
not harvestable and (cheap or sweet)

Set mebership functions:

harvestable : { 0.0 if the fruit is more than 60cm underground,
                1.0 - ((X * X) / 3600) if the fruit is Xcm underground (X < 60),
                1.0 if the fruit is up to 200cm above ground,
                1.0 - ((X - 200) / 100) if the fruit is 200cm to 300cm above ground,
                0.0 if the fruit is more than 300cm above ground }

cheap : { 1.0 if the price per kilogram is less than or equal to J$100,
          100 / PricePerKG if the price per kilogram is more than J$100 )

sweet : { 1.0 if the sugar content is very high,
          0.8 if the sugar content is high,
          0.5 if the sugar content is medium,
          0.2 if the sugar content is low,
          0.0 if there is no sugar content }