# A function 𝐹(𝐴, 𝐵, 𝐶) defined by three Boolean variables A, B and C when expressed as sum of products is given by

Q. A function 𝐹(𝐴, 𝐵, 𝐶) defined by three Boolean variables A, B and C when expressed as sum of products is given by

𝐹 = 𝐴̅ ⋅ 𝐵̅ ⋅ 𝐶̅ + 𝐴̅ ⋅ 𝐵 ⋅ 𝐶̅ + 𝐴 ⋅ 𝐵̅ ⋅ 𝐶̅

where, 𝐴̅, 𝐵̅, and 𝐶̅ are the complements of the respective variables. The product of sums (POS) form of the function F is

(A) 𝐹 = (𝐴 + 𝐵 + 𝐶) ⋅ (𝐴 + 𝐵̅ + 𝐶) ⋅ (𝐴̅ + 𝐵 + 𝐶)

(B) 𝐹 = (𝐴̅ + 𝐵̅ + 𝐶̅) ⋅ (𝐴̅ + 𝐵 + 𝐶̅) ⋅ (𝐴 + 𝐵̅ + 𝐶̅)

(C) 𝐹 = (𝐴 + 𝐵 + 𝐶̅) ⋅ (𝐴 + 𝐵̅ + 𝐶̅) ⋅ (𝐴̅ + 𝐵 + 𝐶̅) ⋅ (𝐴̅ + 𝐵̅ + 𝐶) ⋅ (𝐴̅ + 𝐵̅ + 𝐶̅)

(D) 𝐹 = (𝐴̅ + 𝐵̅ + 𝐶) ⋅ (𝐴̅ + 𝐵 + 𝐶) ⋅ (𝐴 + 𝐵̅ + 𝐶) ⋅ (𝐴 + 𝐵 + 𝐶̅) ⋅ (𝐴 + 𝐵 + 𝐶)

Ans: F = (A + B + C̅) . (A + B̅ + C̅) . (A̅ + B + C̅) . (A̅ + B̅ + C) . (A̅ + B̅ + C̅)

Sol:

F = A̅.B̅.C̅ + A̅.B.C̅ + A.B̅.C̅

In terms of minterms, this can be represented as:

F = ∑m (0, 2, 4)

The equivalent maxterm will contain the terms not present in the minterm representation, i.e.

F = ∑m (0, 2, 4) = π(1, 3, 5, 6, 7) = M1. M3. M5. M6. M7

⇒ (A + B + C̅) (A + B̅ + C̅) (A̅ + B + C̅) (A̅ + B̅ + C) (A̅ + B̅ + C̅)

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