GATE EC 2015 Set 2 — Question 46

The given function is in Sum of Products (SOP) form: F(X, Y, Z) = Σm(1, 2, 5, 6, 7).
To find the Product of Sums (POS) form, we first identify the maxterms, which correspond to the minterms not present in the SOP form.
The total minterms for 3 variables are from 0 to 7.
The missing minterms in the given function are 0, 3, and 4.
So, the function can be represented in POS form as a product of the corresponding maxterms: F(X, Y, Z) = ΠM(0, 3, 4).Let's write the sum terms for these maxterms:

• M₀ corresponds to input 000, so the sum term is (X+Y+Z).
• M₃ corresponds to input 011, so the sum term is (X+$\bar{Y}$+$\bar{Z}$).
• M₄ corresponds to input 100, so the sum term is ($\bar{X}$+Y+Z).

Therefore, the correct POS expression is F(X, Y, Z) = (X+Y+Z)⋅(X+$\bar{Y}$+$\bar{Z}$)⋅($\bar{X}$+Y+Z).Now let's examine the given options:
(A) ($\bar{X}$+$\bar{Y}$+Z)⋅($\bar{X}$+Y+$\bar{Z}$)⋅(X+$\bar{Y}$+Z) corresponds to ΠM(6, 5, 2).
(B) (X+Y+Z)⋅(X+$\bar{Y}$+Z)⋅($\bar{X}$+Y+Z) corresponds to ΠM(0, 2, 4).
(C) This is a product of 5 terms.
(D) This is a product of 5 terms.None of the options match the derived POS expression. There is a known error in this question from the official GATE 2015 paper. The marked correct answer is (B). Option (B) corresponds to the function F = ΠM(0, 2, 4), which is equivalent to F = Σm(1, 3, 5, 6, 7). This differs from the question's function F = Σm(1, 2, 5, 6, 7) by one minterm (m₃ vs m₂). Assuming the intended function was F = Σm(1, 3, 5, 6, 7), option (B) would be correct.