Material nonimplication

Material nonimplication

NIMPLY
Definition	$x{\overline {y}}$
Truth table	$(0100)$
Logic gate
Normal forms
Disjunctive	$x{\overline {y}}$
Conjunctive	$x+{\overline {y}}$
Zhegalkin polynomial	$x\oplus xy$
Post's lattices
0-preserving	yes
1-preserving	no
Monotone	no
Affine	no
Self-dual	no

Material nonimplication or abjunction (from Latin ab 'away' and junctio 'to join') is a term referring to a logic operation used in generic circuits and Boolean algebra. It is the negation of material implication. That is to say that for any two propositions $P$ and $Q$, the material nonimplication from $P$ to $Q$ is true if and only if the negation of the material implication from $P$ to $Q$ is true. This is more naturally stated as that the material nonimplication from $P$ to $Q$ is true only if $P$ is true and $Q$ is false.

It may be written using logical notation as $P\nrightarrow Q$, $P\not \supset Q$, or "Lpq" (in Bocheński notation), and is logically equivalent to $\neg (P\rightarrow Q)$, and $P\land \neg Q$.