Discussion
Consider the following 4-variable Boolean function
\(F(A,B,C,D) = \Sigma m(0,1,2,3,8,9,10,11)\)
Consider A as MSB, D as LSB. Which one of the following options represents the minimal sum of products form for the above function?
Note: + is OR operation, . is AND operation, ′ is NOT operation
Answer: B
The minimal sum of products is B′.
With A as the most significant bit, the minterms 0-3 are 0000-0011 and 8-11 are 1000-1011. In all eight, B = 0, and A, C and D take every combination of values. So F = 1 exactly when B = 0:
\[ F = \Sigma m(0,1,2,3,8,9,10,11) = B' \]
On a Karnaugh map these eight cells form a single group of eight covering the two B′ rows, which reduces to the single literal B′.
A′ would cover minterms 0-7, including 4-7 where B = 1. A′.B′ + A.B covers 0-3 and 12-15, the wrong second half. A′ + B′ + C′ + D′ is 1 for every minterm except 15.
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