A functionally complete operation is an operation that can be used to represent any Boolean function or logical expression. In other words, if an operation is functionally complete, it means that you can create any desired logical expression using only that operation.

Logical NOR (also called Peirce's arrow or the Quine dagger) is a binary operation that takes two input values (A and B) and returns true (1) only if both A and B are false (0); otherwise, it returns false (0). The truth table for NOR looks like this:

A | B | NOR(A, B)
--+---+---------
0 | 0 | 1
0 | 1 | 0
1 | 0 | 0
1 | 1 | 0

Logical NOR is considered functionally complete because it can be used to construct other basic logical operations such as NOT, AND, and OR, which can be further combined to create any Boolean function or logical expression.

Here's how we can derive the basic logical operations using NOR:

NOT (negation):
NOT(A) = NOR(A, A)
Truth table:
A | NOT(A)
--+-------
0 | 1
1 | 0

AND:
AND(A, B) = NOR(NOR(A, A), NOR(B, B))
Truth table:
A | B | AND(A, B)
--+---+---------
0 | 0 | 0
0 | 1 | 0
1 | 0 | 0
1 | 1 | 1

OR:
OR(A, B) = NOR(NOR(A, B), NOR(A, B))
Truth table:
A | B | OR(A, B)
--+---+---------
0 | 0 | 0
0 | 1 | 1
1 | 0 | 1
1 | 1 | 1

Since you can create any Boolean function or logical expression by combining NOT, AND, and OR gates, and you can create these gates using only the NOR operation, this makes NOR a functionally complete operation.
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