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

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

𝐹 = 𝐴̅ β‹… 𝐡̅ β‹… 𝐢̅ + 𝐴̅ β‹… 𝐡 β‹… 𝐢̅ + 𝐴 β‹… 𝐡̅ β‹… 𝐢̅

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

(A) 𝐹 = (𝐴 + 𝐡 + 𝐢) β‹… (𝐴 + 𝐡̅ + 𝐢) β‹… (𝐴̅ + 𝐡 + 𝐢)

(B) 𝐹 = (𝐴̅ + 𝐡̅ + 𝐢̅) β‹… (𝐴̅ + 𝐡 + 𝐢̅) β‹… (𝐴 + 𝐡̅ + 𝐢̅)

(C) 𝐹 = (𝐴 + 𝐡 + 𝐢̅) β‹… (𝐴 + 𝐡̅ + 𝐢̅) β‹… (𝐴̅ + 𝐡 + 𝐢̅) β‹… (𝐴̅ + 𝐡̅ + 𝐢) β‹… (𝐴̅ + 𝐡̅ + 𝐢̅)

(D) 𝐹 = (𝐴̅ + 𝐡̅ + 𝐢) β‹… (𝐴̅ + 𝐡 + 𝐢) β‹… (𝐴 + 𝐡̅ + 𝐢) β‹… (𝐴 + 𝐡 + 𝐢̅) β‹… (𝐴 + 𝐡 + 𝐢)

Ans: F = (A + B + CΜ…) . (A + BΜ… + CΜ…) . (AΜ… + B + CΜ…) . (AΜ… + BΜ… + C) . (AΜ… + BΜ… + CΜ…)

Sol:

F = AΜ….BΜ….CΜ… + AΜ….B.CΜ… + A.BΜ….CΜ…

In terms of minterms, this can be represented as:

F = βˆ‘mΒ (0, 2, 4)

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

F =Β βˆ‘m (0, 2, 4) = Ο€(1, 3, 5, 6, 7) = M1. M3. M5. M6. M7

β‡’ (A + B + CΜ…) (A + BΜ… + CΜ…) (AΜ… + B + CΜ…) (AΜ… + BΜ… + C) (AΜ… + BΜ… + CΜ…)

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