Math Problem Statement
Solution
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:
- NOT
- AND
- OR
Step 1: Expressing NOT using NOR
For any Boolean variable , we can write:
Thus, a single NOR operation can implement the NOT operation.
Step 2: Expressing AND using NOR
For two variables and , the AND operation can be expressed as:
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:
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:
- What is a universal gate, and why is it important in Boolean algebra?
- How can the NAND gate also be shown to be functionally complete?
- What is the difference between functional completeness and functional independence?
- Can you implement other Boolean operations like XOR using only {NOR}?
- 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.
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Math Problem Analysis
Mathematical Concepts
Boolean Algebra
Logic Gates
Functional Completeness
Formulas
NOT(A) = A NOR A
AND(A, B) = NOT(A NOR B)
OR(A, B) = NOT(NOT(A) AND NOT(B))
Theorems
Functional Completeness of NOR Gate
Suitable Grade Level
University Level - Logic and Computing Systems