Question

The minimal SOP expression for a 3-variable function having \(m\) in terms of \(\sum m(1,3,5,7)\) is

• A. \(C \)
• B. \(A+C \)
• C. \(B \)
• D. \(AB+C \)

Hint:

Notice the binary pattern for minterms \((1, 3, 5, 7)\): \(001, 011, 101, 111\). The Least Significant Bit (C) is always 1 while A and B toggle through all combinations, meaning the expression simplifies directly to \(C\).

Answer 1

The Correct Option is A

Concept: In digital logic design, a Boolean function can be simplified into its minimal Sum of Products (SOP) form using algebraic identities or standard Karnaugh Map (K-Map) optimization techniques. The question specifies a 3-variable Boolean function. Let us define these three standard binary inputs as \(A\) (Most Significant Bit), \(B\), and \(C\) (Least Significant Bit). The given minterms can be written out in their binary equivalents as follows:
• \(m_1 = \overline{A}\overline{B}C \rightarrow 001_2\)
• \(m_3 = \overline{A}BC \rightarrow 011_2\)
• \(m_5 = A\overline{B}C \rightarrow 101_2\)
• \(m_7 = ABC \rightarrow 111_2\)

Step 1: Expanding the full SOP expression.
We combine the individual minterms using Boolean addition to write the unsimplified function: \[ F(A,B,C) = \sum m(1,3,5,7) = \overline{A}\overline{B}C + \overline{A}BC + A\overline{B}C + ABC \]

Step 2: Applying Boolean algebra theorems to simplify the expression.
We can simplify this expression step-by-step by factoring out common terms. First, group the first two terms together, and group the last two terms together: \[ F = \overline{A}C(\overline{B} + B) + AC(\overline{B} + B) \] Using the Boolean complement law, we know that \(\overline{B} + B = 1\). Substituting this identity into our expression gives: \[ F = \overline{A}C(1) + AC(1) = \overline{A}C + AC \] Now, factor out the common variable \(C\) from the remaining two terms: \[ F = C(\overline{A} + A) \] Applying the complement law again, we know that \(\overline{A} + A = 1\): \[ F = C(1) = C \] Evaluating this result using a standard K-Map shows that all four minterms combine into a single 4-cell group (a quad) that eliminates variables \(A\) and \(B\), leaving only \(C\). Therefore, the minimal Sum of Products expression is simply \(C\), which matches Option (A).