Editing Logical NOR (section)

==Properties==
NOR is commutative but not associative, which means that <math>P \downarrow Q \leftrightarrow Q \downarrow P</math> but <math>(P \downarrow Q) \downarrow R \not\leftrightarrow P \downarrow (Q \downarrow R)</math>.<ref>{{Cite book |last=Rao |first=G. Shanker |url=https://books.google.com/books?id=M-5m_EdvxuIC |title=Mathematical Foundations of Computer Science |date=2006 |publisher=I. K. International Pvt Ltd |isbn=978-81-88237-49-4 |pages=22 |language=en}}</ref>

===Functional completeness===
The logical NOR, taken by itself, is a [[Functional completeness|functionally complete]] set of connectives.<ref name=":29">{{Cite book |last=Smullyan |first=Raymond M. |title=First-order logic |date=1995 |publisher=Dover |isbn=978-0-486-68370-6 |location=New York |pages=5, 11, 14 |language=en}}</ref> This can be proved by first showing, with a [[truth table]], that <math>\neg A</math> is truth-functionally equivalent to <math>A \downarrow A</math>.<ref name=":132">{{Cite book |last=Howson |first=Colin |title=Logic with trees: an introduction to symbolic logic |date=1997 |publisher=Routledge |isbn=978-0-415-13342-5 |location=London; New York |pages=41–43}}</ref> Then, since <math>A \downarrow B</math> is truth-functionally equivalent to <math>\neg (A \lor B)</math>,<ref name=":132" /> and <math>A \lor B</math> is equivalent to <math>\neg(\neg A \land \neg B)</math>,<ref name=":132" /> the logical NOR suffices to define the set of connectives <math>\{\land, \lor, \neg\}</math>,<ref name=":132" /> which is shown to be truth-functionally complete by the [[Disjunctive Normal Form Theorem]].<ref name=":132" />

This may also be seen from the fact that Logical NOR does not possess any of the five qualities (truth-preserving, false-preserving, [[Linear#Boolean functions|linear]], [[Monotonic#In Boolean functions|monotonic]], self-dual) required to be absent from at least one member of a set of [[functionally complete]] operators.