The logical NOR or joint denial is a boolean logic operator which produces a result that is the inverse of logical or. That is, (not or), p NOR q is only true when both p and q are false.

The NOR 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 NOR. Thus, as with NAND operator, NOR can be used by itself, without any other logical operator, to constitute a logical formal system (making NOR functionally complete).

Definition

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The NOR 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 NOR q (also written as p ⊥ q or p ↓ q) is as follows:

Truth Table: Logical NOR

p	q	p ↓ q
F	F	T
F	T	F
T	F	F
T	T	F

One way of expressing p NOR q is $\backslash overline\{p\; \backslash lor\; q\}$, where the symbol $\backslash or$ signifies OR and the bar over the expression signifies the negation of the expression under the bar.

Joint Denial

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NOR has the interesting feature that all other logical operators can be expressed by various functions of NOR.

"not p" is equivalent to "p NOR p"	$\backslash overline\{p\}\; \backslash equiv\; \backslash overline\{p\; +\; p\}$
"p and q" is equivalent to "(p NOR p) NOR (q NOR q)"	$p\; \backslash cdot\; q\; \backslash equiv\; \backslash overline\{\backslash overline\{(p\; +\; p)\}\; +\; \backslash overline\{(q\; +\; q)\}\}$
"p or q" is equivalent to "(p NOR q) NOR (p NOR q)"	$p\; +\; q\; \backslash equiv\; \backslash overline\{\backslash overline\{(p\; +\; q)\}\; +\; \backslash overline\{(p\; +\; q)\}\}$
"p implies q" is equivalent to "((p NOR q) NOR q) NOR ((p NOR q) NOR q)"	$p\; \backslash rightarrow\; q\; \backslash equiv\; \backslash overline\{\backslash overline\{(\backslash overline\{(p+q)\}\; +\; q)\}+\backslash overline\{(\backslash overline\{(p\; +\; q)\}\; +\; q)\}\}$

The logical NAND operator 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

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Other operators

• Exclusive disjunction
• Logical conjunction
• Logical disjunction

• Logical equality
• Logical implication
• Logical NAND

Related topics

• Ampheck
• Boolean algebra
• Boolean domain
• Boolean function

• Boolean logic
• Propositional logic
• Sole sufficient operator
• Zeroth order logic

Logic | Boolean algebra | Binary operations

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