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Material nonimplication

NIMPLY
Definition	$x\bar{y}$
Truth table	$\left(0100\right)$
Logic gate
Normal forms
Disjunctive	$x\bar{y}$
Conjunctive	$x+\bar{y}$
Zhegalkin polynomial	$x\oplus xy$
Post's lattices
0-preserving	yes
1-preserving	no
Monotone	no
Affine	no
Self-dual	no
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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.^[1] 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 $P\nrightarrow Q$, $P\supset ̸Q$, or "Lpq" (in Bocheński notation), and is logically equivalent to $\neg (P\to Q)$, and $P\wedge \neg Q$.

falsehood-preserving: The interpretation under which all variables are assigned a truth value of "false" produces a truth value of "false" as a result of material nonimplication.

The symbol for material nonimplication is simply a crossed-out material implication symbol. Its Unicode symbol is 219B₁₆ (8603 decimal): ↛.

Bitwise operation: P & ~Q. This is usually called "bit clear" (BIC) or "and not" (ANDN).

Logical operation: P && !Q.

• Implication
• Set difference

• Media related to Material nonimplication at Wikimedia Commons

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