Exercise 2.1

1. Explain why the size of the hypothesis space in the EnjoySport learning task is 973.
   
   The hypothesis space cardinality is the number of different combinations of the possible values for each attribute represented in thehypothesis space. The values in the hypothesis space for each attribute include the specific values that can be assigned in addition to the "?" (any value is acceptable) symbol and "Ø" (no value is acceptable) symbol. Since the "Ø" symbol for any attribute always classifies an instance as negative, we count all the hypotheses with this value only once. Consider it the "never" hypothesis.
   
   

   Attributes:	Sky		AirTemp		Humidity		Wind		Water		Forecast		Total
   # Values:	3	*	2	*	2	*	2	*	2	*	2	=	96
   + 1 ("?")	4	*	3	*	3	*	3	*	3	*	3	=	972
   + 1 for the "never" hypothesis												+	1
   Total:													973

   ---

2. How would the number of possible instances and possible hypotheses increase with the addition of the attribute WaterCurrent, which can take on the values Light, Moderate, or Strong?
   
   Instances: All that has to be done is to factor in 3 for values the new attribute can assume in the calculation we did before.
   
   

   3² * 2⁵ = 288 instances.

   Possible Hypotheses: All that has to be done is to factor in 4 for values the new attribute can assume plus the "?" symbol in the calculation we did before.
   
   

   4² * 3⁵ = 3889 possible hypotheses.

   ---

3. More generally, how does the number of possible instances and hypotheses grow with the addition of a new attribute A that takes on k possible values?
   
   Instances: The new number of instances is simply:
   
   

   current # * k

   Possible Hypotheses: This is a little more complicated, but not much.
   
   

   ( current # * ( k + 1 ) ) - k

   The ( k + 1 ) takes into account the "?" symbol. The ( - k ) is for the extra times the "never" hypothesis was counted by multiplying by ( k + 1 ).

   ---