Consider the following four variable Boolean function in sum-of-product form…
2025
Consider the following four variable Boolean function in sum-of-product form
\(𝐹(𝑏_3, 𝑏_2, 𝑏_1, 𝑏_0) = ∑(0, 2, 4, 8, 10, 11, 12).\)
where the value of the function is computed by considering \(𝑏_3𝑏_2𝑏_1𝑏_0\) as a 4-bit binary number, where \(𝑏_3\) denotes the most significant bit and \(𝑏_0\) denotes the least significant bit. Note that there are no don’t care terms. Which ONE of the following options is the CORRECT minimized Boolean expression for \(F\)?
Answer: A. \(\bar{b}_1 \bar{b}_0 + \bar{b}_2 \bar{b}_0 + b_1 \bar{b}_2 b_3 \) — Minimized expression: b1' b0' + b2' b0' + b1 b3 b2' Derivation (grouping by covered minterms): b1' b0' covers minterms 0 (0000), 4 (0100), 8 (1000), and 12…
- A.
\(\bar{b}_1 \bar{b}_0 + \bar{b}_2 \bar{b}_0 + b_1 \bar{b}_2 b_3 \) - B.
\(\bar{b}_1 \bar{b}_0 + \bar{b}_2 \bar{b}_0 \) - C.
\(\bar{b}_2 \bar{b}_0 + b_1 b_2 b_3 \) - D.
\(\bar{b}_0 \bar{b}_2 + \bar{b}_3\)
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Correct answer: A
Minimized expression: b1' b0' + b2' b0' + b1 b3 b2'
Derivation (grouping by covered minterms):
b1' b0' covers minterms 0 (0000), 4 (0100), 8 (1000), and 12 (1100).
b2' b0' covers minterms 0 (0000), 2 (0010), 8 (1000), and 10 (1010).
b1 b3 b2' covers minterms 10 (1010) and 11 (1011); minterm 10 is already covered, and this term adds the required minterm 11.
Combine coverage:
Union of covered minterms = {0, 2, 4, 8, 10, 11, 12}, exactly the given set.
No extra minterms are covered, so the expression matches the function.
Therefore the minimized Boolean expression is b1' b0' + b2' b0' + b1 b3 b2'.
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