For the following questions, use the traditional definition of fuzzy subsethood.

1. "Very" is often used as a modifier to reduce vagueness on fuzzy truth functions and set membership.  The usual interpretation is that if  statement "A is true" has truth value x, then the statement "A is very true" has truth value x²  .  The rationale behind this is that being "very true" is a more demanding requirement.
   For example, if "The water is clean" has truth value 0.7, then "The water is very clean" has truth value 0.49.
   
   True or false (and explain):  Let "S" be a fuzzy set.  Then "Very S" is a fuzzy subset of "S".
   
   
2. Similarly, "more or less" is used as a modifier to increase vagueness - the usual interpretation is that if "A is true" has truth value x, then "A is more or less true" has truth value SQRT(x).
   
   Example:  if "The students are happy" has truth value  0.25, then "The students are more or less happy" has truth value 0.5
   
   True or false (and explain):  Let "S" be a fuzzy set.  Then "S" is a fuzzy subset of "more or less S"
   
   
3. Using the definitions just given, is it true that "not very S" is a subset of "more or less S", or vice versa, or is it impossible to predict?
   
   
4. Is "not more or less S" a subset of "very S", or vice versa, or is it impossible to predict?
   
   
5. Suppose A(x) has the truth function 
   A(x) = 1 for x <= 2
   A(x) = 1 - (x-2)/3      for 2 < x < 5
   A(x) = 0 for x >= 5
   
   and B(x) has the truth function
   B(x) = 0 for x <= 3
   B(x) = (x-3)/4            for 3 < x < 7
   B(x) = 1 for x >= 7
   
   Find the value x that has the largest possible truth value for "not (A(x) OR B(x))"