Let's simplify the given expression step by step to determine which of the options is equivalent to the given Boolean expression.

The expression is:

$A \cdot B \cdot C + A \cdot B \cdot C \cdot (\overline{D} + E) \cdot (D + C)$

Step 1: Factorization of terms

1. From the given expression, the first term is: $A \cdot B \cdot C$ The second term is: $A \cdot B \cdot C \cdot (\overline{D} + E) \cdot (D + C)$

Step 2: Simplification of the second term

• Expand $(\overline{D} + E) \cdot (D + C)$ using the distributive property: $(\overline{D} + E) \cdot (D + C) = \overline{D} \cdot D + \overline{D} \cdot C + E \cdot D + E \cdot C$

• Simplify: $\overline{D} \cdot D = 0, \quad \text{so the expression becomes:}$ $\overline{D} \cdot C + E \cdot D + E \cdot C$

Step 3: Substitute back into the original expression

The original expression becomes: $A \cdot B \cdot C + A \cdot B \cdot C \cdot (\overline{D} \cdot C + E \cdot D + E \cdot C)$

Step 4: Factor out $A \cdot B \cdot C$

Factor $A \cdot B \cdot C$ from both terms: $A \cdot B \cdot C \cdot \left(1 + \overline{D} \cdot C + E \cdot D + E \cdot C\right)$

• Since $1 + X = 1$ for any Boolean expression $X$, this simplifies to: $A \cdot B \cdot C$

Final Expression:

The equivalent expression is simply: $A \cdot B \cdot C$

Correct Option:

The correct answer is $A \cdot B \cdot C$.

Would you like a breakdown of this process or further clarifications?

Related Questions:

1. How do you apply Boolean laws to simplify complex expressions?
2. What is the role of distributive property in Boolean algebra?
3. Can DeMorgan’s laws help simplify Boolean expressions?
4. How does factoring simplify Boolean equations in logic circuits?
5. What are some practical applications of Boolean simplifications in computing?

Tip:

Always look for opportunities to simplify using fundamental Boolean laws, such as the Idempotent Law ($X + X = X$) and the Complement Law ($X \cdot \overline{X} = 0$).