What is the minimal SOP form for F = Σm(0,1,2,3,8,9,10,11) in GATE CS 2026 Set-2 Q40?

The minimal sum-of-products form is B'' (Option B). Converting the minterms to binary with A as MSB: 0=0000, 1=0001, 2=0010, 3=0011, 8=1000, 9=1001, 10=1010, 11=1011. All minterms have B=0 (second bit is 0). The variables A, C, D take all possible combinations while B is always 0, so F = B''.

How do you identify patterns in a minterm list for Boolean minimization?

Convert each minterm to binary and look for common bit patterns. If a particular bit position has the same value (0 or 1) across all minterms, that variable appears in its literal form (or complement) in the minimal expression. If a bit varies freely across minterms while others are constant, that variable does not appear in the final simplified form. This method quickly identifies single-literal or two-literal minimal forms.

Why is B'' alone the minimal form and not B''·C'' or another expression?

For 4 variables (A,B,C,D), when B=0 there are 2 to the power 3 = 8 possible combinations of the remaining variables A, C, D. These correspond exactly to minterms 0,1,2,3,8,9,10,11 — which matches the given minterm list completely. Since all 8 combinations are present, no restriction on A, C, or D exists, so they do not appear in the expression. Only B=0 is required, giving F = B''.

Boolean Function Minimal SOP Form B Complement | GATE CS 2026 Set-2 Q40

Computer Sciences > GATE 2026 SET-2 > Digital Logic

Consider the following 4-variable Boolean function
F(A, B, C, D) = Σm(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

A' + B' + C' + D'

A'.B' + A.B

Correct : b

The correct answer is Option B — B''''.
The function F(A, B, C, D) = Σm(0,1,2,3,8,9,10,11) with A as MSB and D as LSB can be analysed by converting each minterm to its binary representation. The minterms are: m0=0000, m1=0001, m2=0010, m3=0011, m8=1000, m9=1001, m10=1010, m11=1011.
Pattern recognition: All minterms have B = 0 (the second bit is 0 in all cases). The values of A, C, and D vary freely among these minterms — A is either 0 or 1, C is either 0 or 1, and D is either 0 or 1. The only constant is B = 0.
Verification: B'''' (B = 0) covers exactly minterms where the second bit is 0, which includes all 8 minterms: 0,1,2,3,8,9,10,11 (matches the given set exactly) and also 4,5,6,7,12,13,14,15... wait, that''''s 16 minterms total for B=0.
Actually — let me recount. For 4 variables with B=0, there are 2³ = 8 combinations of (A,C,D). These 8 minterms are exactly 0,1,2,3,8,9,10,11 ✓.
Therefore the minimal sum-of-products form is simply F = B''''.

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