self doubt

Question

For given assignment of truth table value, number of min terms will evaluate to 1:

a)1/2 of all (atleast)

b)1/2 of all (atmost)

c)1/4 of all (atleast)

d)1/4 of all (atmost)

Answer 1

Let the number of variables be n.

For n = 1,
      for a given assignment, there will always be exactly 1 minterm which would evaluate to 1. Hence out of 2 minterms, exactly 1 is satisfying the condition. $\Rightarrow \frac{1}{2}$
eg. for f(A) and given assignment A=1, we have only one minterm resulting in a 1 ie A    
OR if given assignment is A=0 then also exactly 1 minterm resulting in 1 ie ~A

Similarly for n = 2
     Exactly 1 min term out of 4 would evaluate to 1. $\Rightarrow \frac{1}{4}$
 

Generalizing,
   For n variables, we would have 1 out of $2^n$ minterms evaluating to 1 $\Rightarrow \frac{1}{2^n}$

We achieved the maximum at n = 1 hence answer is $\frac{1}{2}$ (atmost)