Expert: Ahmed Salami - 12/14/2009

Question
What percent of the time is the following statement True?

(A ∧ ~ B ) ∨ ~ C  

Answer
Hi Ashley,
We make a table for all the possible combinations of A, B and C and see the ones that make the statement true.
A  B  C  [(A ∧ ~B) ∨ ~C]
F  F  F          T
F  F  T          F
F  T  F          T
F  T  T          F
T  F  F          T
T  F  T          T
T  T  F          T
T  T  T          F
And so we see that (A ∧ ~ B ) ∨ ~ C is true 5 out of 8 times
i.e 5/8 x 100 = 62.5%

Or we could have just looked at it logically and said; the statement is always true when ~C is true (or C is false) which happens 50% of the time. Of the remaining 50% (when C is true) the statement is true when (A ∧ ~B) is true i.e when A is true and B is false, and this only happens once in the four times. So, the statement is true for
50% + (1/4)50% = 50% + 12.5%
              = 62.5%

Regards