The function $F(A, B, C) = A(A + \overline{C})(A + B)$ represents a Boolean logic expression. Let's simplify and analyze it step by step.

Step 1: Expand the Expression

We will expand $F(A, B, C)$ using Boolean algebra rules:

$F(A, B, C) = A \cdot (A + \overline{C}) \cdot (A + B)$

Distributive Property

• $A \cdot (A + \overline{C}) = A$ because $A + \overline{C}$ always includes $A$, and $A \cdot A = A$ (idempotent law).

Thus, the expression simplifies to:

$F(A, B, C) = A \cdot (A + B)$

Distributive Property Again

• $A \cdot (A + B) = A$ because $A + B$ always includes $A$.

Final Simplified Expression

$F(A, B, C) = A$

Explanation

The final result shows that the function $F(A, B, C)$ always outputs $A$, regardless of the values of $B$ or $C$. This simplification demonstrates that the logic gates in the original expression redundantly depend on $A$.

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Would you like a more detailed explanation of the simplification process or its implications? Here are 5 related questions to expand the topic:

1. What are the basic laws of Boolean algebra used in this simplification?
2. How can this function be implemented using logic gates?
3. What happens if we replace $\overline{C}$ with $C$ in the original equation?
4. How does the simplification process differ when other variables (like $B$ or $C$) are dominant?
5. Can this Boolean function be represented using a truth table?

Tip:

When simplifying Boolean expressions, always start by applying the idempotent, distributive, and absorption laws systematically.