The task given in the image is:

"Zeigen Sie, dass {V̅} funktional vollständig ist."
(Translation: "Show that {V̅} is functionally complete.")

Functional Completeness

In logic and Boolean algebra, a set of logical operators is functionally complete if it can be used to express all possible Boolean functions.

• The overlined "V̅" likely represents the logical operation NOR (Negated OR).
• The NOR gate is an important operator because it is a universal gate, meaning it can construct all other basic logical operations, like AND, OR, and NOT.

Proof Outline:

To prove that {NOR} (or {V̅}) is functionally complete, we need to show that it can represent the fundamental Boolean operations:

1. NOT
2. AND
3. OR

Step 1: Expressing NOT using NOR

For any Boolean variable $A$, we can write:
$\text{NOT}(A) = A \, \text{NOR} \, A = \overline{A \vee A}.$
Thus, a single NOR operation can implement the NOT operation.

Step 2: Expressing AND using NOR

For two variables $A$ and $B$, the AND operation can be expressed as:
$A \, \text{AND} \, B = \text{NOT} \left( A \, \text{NOR} \, B \right).$
By Step 1, we already know how to implement NOT using NOR.

Step 3: Expressing OR using NOR

The OR operation can be derived using De Morgan's law:
$A \, \text{OR} \, B = \text{NOT} \left( \text{NOT}(A) \, \text{AND} \, \text{NOT}(B) \right).$
Since we have NOT and AND expressed using NOR, OR can also be expressed.

Conclusion:

Since we can express NOT, AND, and OR using the NOR operator, the set {V̅} (NOR) is functionally complete.

Let me know if you would like a detailed step-by-step breakdown or further clarification!

Related Questions:

1. What is a universal gate, and why is it important in Boolean algebra?
2. How can the NAND gate also be shown to be functionally complete?
3. What is the difference between functional completeness and functional independence?
4. Can you implement other Boolean operations like XOR using only {NOR}?
5. How does the NOR gate differ in practical applications compared to the NAND gate?

Tip: Always verify functional completeness by expressing all three key operations (NOT, AND, OR) explicitly.