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Logical NOR

The logical NOR is an operation on two logical values, typically the values of two propositions, that produces a value of true if both of its operands are false. In other words, it produces a value of false if at least one of its operands is true. ↓ is also known as the Peirce arrow after its inventor, Charles Sanders Peirce, and is a Sole sufficient operator.

The truth table for p NOR q (also written as p ↓ q, or Xpq) is as follows:

The negation of a disjunction ¬(p ∨ q), and the conjunction of negations (¬ p) ∧ (¬ q) can be tabulated as follows:

Inspection of the tabular derivations for NAND and NOR, under each assignment of logical values to the functional arguments p and q, produces the identical patterns of functional values for ¬(p ∧ q) as for (¬ p) ∨ (¬ q), and for ¬(p ∨ q) as for (¬ p) ∧ (¬ q). Thus the first and second expressions in each pair are logically equivalent, and may be substituted for each other in all contexts that pertain solely to their logical values.

This equivalence is one of De Morgan's laws.