Question

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. 𝐴′ + 𝐵′ + 𝐶′ + 𝐷′
• B. 𝐵′
• C. 𝐴′. 𝐵′ + 𝐴 . 𝐵
• D. 𝐴′

Hint:

All eight given minterms have \(B=0\), while \(A\), \(C\), \(D\) vary freely, so the function simplifies directly to the single literal \(B'\).

Answer 1

The Correct Option is B

Step 1: Write out the 4-bit binary code \(ABCD\) for each minterm index: \(0=0000\), \(1=0001\), \(2=0010\), \(3=0011\), \(8=1000\), \(9=1001\), \(10=1010\), \(11=1011\).
Step 2: Scan the \(B\) column across all eight entries -- in every single one, \(B\) equals \(0\). The variable \(A\) switches between \(0\) (for minterms 0-3) and \(1\) (for minterms 8-11), and \(C,D\) run through all four combinations \(00,01,10,11\) in both halves.
Step 3: Because every combination of \(A\), \(C\), \(D\) appears alongside \(B=0\), and none of the eight minterms with \(B=1\) are present in the sum, the function is exactly equivalent to the single condition \(B=0\).
Step 4: On a K-map, this shows up as one clean rectangular block of 8 adjacent cells spanning the entire \(B=0\) half of the map -- the largest possible power-of-2 grouping in a 4-variable map -- which reduces to just the literal \(B'\), since \(A\), \(C\), and \(D\) all vary freely within that block and therefore drop out.
Step 5: No smaller sum-of-products expression can represent this function, since a single-literal term is already the simplest non-trivial form.
Final Answer: \(B'\)