1. Let a be a crisp number.  We might define a truth value for the fuzzy statement "x <= a" as follows:
• value = 1                      if x <= a (using the normal definition of <=)
  
• value =  (1 + a - x)       if a <= x <= a+1
• value = 0                      if x > a+1


Draw the truth function for the fuzzy statement "x <= 10"


2. Let a be a crisp number.  We might define a truth value for the fuzzy statement "x >= a" as follows:
• value = 0                       if x < a-1
  
• value =  (1 + x - a)        if a-1 <= x <= a
• value = 1                       if x >= a

Draw the truth function for the fuzzy statement "x >= 5"


3. Using the definitions just given, sketch the truth functions for each of the following fuzzy statements
1. "x is <= 5 and x is >= 6"
2. "x is <= 5 or x is >= 6"
3. "x is >= 2 and x is <= 4"
4. "x is >= 2 or x is <= 4"
5. "x is >= 4 and x is <= 4"
   
   
4. How does part 5. of Exercise 3. relate to the concept of numeric equality in fuzzy logic?
   
   
5. Let P(x) and Q(x) be fuzzy truth functions, each of which can only give truth values of 0, 0.5, and 1.  That is, for all x P(x) is in the set {0, 0.5, 1} and Q(x) is in the set {0, 0.5, 1}
   
   Recall the Kleene-Dienes definition of implication:  "a -> b" is equivalent to "(not a) or b"
   
   Compute the truth table for the fuzzy statement "(P(x) and (P(x) -> Q(x))) -> Q(x)"
   
   How does this compare for the same truth table in crisp logic (where P(x) and Q(x) can only be "true" or "false")?