The given Boolean function is:

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

Let’s simplify this step by step.

Step 1: Combine common terms

$\overline{C} + A + B + \overline{C} = \overline{C} + A + B$

This is because $\overline{C} + \overline{C} = \overline{C}$.

So the function becomes:

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

Step 2: Simplify using Boolean properties

• Apply distribution of terms and check for simplifications.
• $(A + B + C)$ is common with other terms and will help further reduction.

I will now perform the detailed Boolean simplification.The simplified Boolean expression is:

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

This means the function $F$ can be reduced to the logical OR of $A$ and the AND of $B$ and $C$.

Do you have any questions or need further details on the steps?

Here are some related questions for further exploration:

1. How does the distributive property apply in Boolean algebra?
2. What are the common Boolean identities used in simplification?
3. Can this expression be simplified further if additional constraints are applied?
4. How would the function behave if we assign specific values to $A$, $B$, and $C$?
5. How is this simplification helpful in digital circuit design?

Tip: In Boolean algebra, always look for common factors and apply identities like De Morgan's law to simplify complex expressions efficiently.