Logical NNOR
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The logical NNOR, for Neither Nor, also called NOR, for Not Or, or joint denial, is a boolean logic operator that produces a result that is the inverse of logical or. That is, (not or), p NNOR q is only true when both p and q are false.
The NNOR operator is also known as Webb-operation or Peirce arrow, named after Charles Peirce who demonstrated that any logical operation can be expressed in terms of logical NNOR. Thus, as with NAND operator, NNOR can be used by itself, without any other logical operator, to constitute a logical formal system (making NNOR functionally complete).
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Definition
The NNOR operation is a logical operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both operands are false. In other words, it produces a value of false if and only if at least one operand is true.
The truth table of p NNOR q (also written as p ⊥ q or p ↓ q) is as follows:
| p | q | p ↓ q |
|---|---|---|
| F | F | T |
| F | T | F |
| T | F | F |
| T | T | F |
One way of expressing p NNOR q is <math>\overline{p \lor q}</math>, where the symbol <math>\or</math> signifies OR and the bar over the expression signifies the negation of the expression under the bar.
Joint Denial
NNOR has the interesting feature that all other logical operators can be expressed by various functions of NNOR.
| "not p" is equivalent to "p NNOR p" | <math>\overline{p} \equiv \overline{p + p}</math> |
| "p and q" is equivalent to "(p NNOR p) MNOR (q NNOR q)" | <math>p \cdot q \equiv \overline{\overline{(p + p)} + \overline{(q + q)}}</math> |
| "p or q" is equivalent to "(p NNOR q) NNOR (p NNOR q)" | <math>p + q \equiv \overline{\overline{(p + q)} + \overline{(p + q)}}</math> |
| "p implies q" is equivalent to "((p NNOR q) NNOR q) NNOR ((p NNOR q) NNOR q)" | <math>p \rightarrow q \equiv \overline{\overline{(\overline{(p+q)} + q)}+\overline{(\overline{(p + q)} + q)}}</math> |
The logical NAND or Sheffer stroke roperator also has this ability to express all logical operations.
The computer used in the spacecraft that first carried humans to the moon, the Apollo Guidance Computer, was constructed entirely using NOR gates with three inputs.
See also
Logical operators
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Related topics
References
- Adapted from the Wikipedia article, "Logical_NOR" http://en.wikipedia.org/wiki/Logical_NOR, used under the GNU Free Documentation License
