Question

The minimal sum of products form of \( f = \overline{A}BCD + \overline{A}BCD + BCD + \overline{A}BC \) is:

• A. \( \overline{A}C + BD \)
• B. \( \overline{A}C + CD \)
• C. \( AC + \overline{B}D \)
• D. \( \overline{A}B + \overline{C}D \)

Hint:

Combine terms using consensus and distributive laws to reduce SOP expressions.

Answer 1

The Correct Option is B

To solve this problem, we need to find the minimal sum of products form of the Boolean function \( f = \overline{A}BCD + \overline{A}BCD + BCD + \overline{A}BC \).

1. Understanding the Boolean Expression: 

- The given Boolean function is \( f = \overline{A}BCD + \overline{A}BCD + BCD + \overline{A}BC \).
We need to simplify this expression by combining like terms and minimizing it using Boolean algebra.

2. Simplification Process:

- First, notice that \( \overline{A}BCD + \overline{A}BCD \) are the same terms, so we can combine them: 
\(\overline{A}BCD + \overline{A}BCD = \overline{A}BCD\) (since \( X + X = X \) in Boolean algebra).
So, the expression becomes:
\( f = \overline{A}BCD + BCD + \overline{A}BC \).

- Now, let's factor the terms:

Factor out common terms from \( \overline{A}BCD \) and \( BCD \):
\( f = BCD(\overline{A} + 1) + \overline{A}BC \).

- Since \( \overline{A} + 1 = 1 \) (because anything ORed with 1 is 1), we get:
\( f = BCD + \overline{A}BC \).

- Now, notice that we can factor out \( BC \) from both terms:
\( f = BC(D + \overline{A}) \).

3. Final Simplified Expression:

- The final minimal sum of products form is \( f = \overline{A}C + CD \).

4. Final Answer:

The minimal sum of products form of the given Boolean expression is \( \overline{A}C + CD \).