GATE CS 2024 Set 1 — Question 47

Given the Boolean function in Sum of Minterms (SOP) form:$$F(X, Y, Z) = \sum(3, 5, 6, 7)$$Analysis of Option (A):
The function is defined over 3 variables, so the total minterms range from 0 to 7. The maxterms correspond to the minterms that are not present in the function.
Missing minterms: $\{0, 1, 2, 4\}$.
Thus, the Product of Maxterms (POS) form is:$$F(X, Y, Z) = \Pi(0, 1, 2, 4)$$This statement is CORRECT.Analysis of Option (B):
Let's simplify the function using the minterms:

• $m_3 = 011$ ($\bar{X}YZ$)
• $m_5 = 101$ ($X\bar{Y}Z$)
• $m_6 = 110$ ($XY\bar{Z}$)
• $m_7 = 111$ ($XYZ$)

We can group them:

• Group $(m_3, m_7)$: $\bar{X}YZ + XYZ = YZ(\bar{X} + X) = YZ$
• Group $(m_5, m_7)$: $X\bar{Y}Z + XYZ = XZ(\bar{Y} + Y) = XZ$
• Group $(m_6, m_7)$: $XY\bar{Z} + XYZ = XY(\bar{Z} + Z) = XY$

Combining these terms:$$F(X, Y, Z) = XY + YZ + XZ$$This is the majority function, which is 1 if two or more inputs are 1. The minterms 3, 5, 6, 7 are exactly those with two or more 1s.
This statement is CORRECT.Analysis of Options (C) and (D):
The expression $F = XY + YZ + XZ$ depends on all three variables. For instance:

• If $X=0, Z=1$, then $F = Y(1) + 0 + 0 = Y$. The output depends on $Y$.
• If $Y=0, Z=1$, then $F = 0 + 0 + X(1) = X$. The output depends on $X$.

Thus, the function is not independent of $X$ or $Y$.
These statements are INCORRECT.