The problem provided asks to show that for a Boolean function $F$ represented in the variables $x_1, x_2, \dots, x_n$, the complement of $F$, denoted $F^d(x_1, \dots, x_n)$, is equal to $F(\overline{x_1}, \dots, \overline{x_n})$.

This is a standard result in Boolean algebra known as De Morgan's Law, generalized for Boolean functions. The steps involve:

Approach:

1. Definition of $F^d(x_1, \dots, x_n)$:

   • By definition, $F^d$ is the complement of the Boolean function $F(x_1, \dots, x_n)$. This means: $F^d(x_1, \dots, x_n) = \overline{F(x_1, \dots, x_n)}$

2. Applying De Morgan's Laws:

   • De Morgan's laws apply to Boolean operations, particularly to AND and OR. The laws state: $\overline{A \cdot B} = \overline{A} + \overline{B} \quad \text{and} \quad \overline{A + B} = \overline{A} \cdot \overline{B}$
   • When generalized for functions, these laws imply that complementing a function negates the inputs to the individual variables. That is, complementing $F(x_1, \dots, x_n)$ can be expressed as: $\overline{F(x_1, \dots, x_n)} = F(\overline{x_1}, \dots, \overline{x_n})$

3. Conclusion:

   • Thus, by De Morgan's Law applied to Boolean functions, we conclude: $F^d(x_1, \dots, x_n) = F(\overline{x_1}, \dots, \overline{x_n})$

This demonstrates the required result.

Would you like further clarification or details on this solution?

Here are 5 related questions to expand your understanding:

1. What are De Morgan's laws and how are they applied in Boolean algebra?
2. How does complementing individual variables affect the entire Boolean expression?
3. Can you derive De Morgan's laws from the basic truth tables of AND, OR, and NOT operations?
4. How does this result generalize to more complex Boolean functions beyond simple conjunctions and disjunctions?
5. What is the significance of the complement of a Boolean function in logic circuit design?

Tip: Always verify Boolean identities using truth tables to gain an intuitive understanding of how they operate across different inputs.