Material nonimplication

Material nonimplication or abjunction (Latin ab = "away", 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 $$P \land \neg Q$$.

Logical Equivalences
Material nonimplication may be defined as the negation of material implication.

In classical logic, it is also equivalent to the negation of the disjunction of $$\neg P$$ and $$Q$$, and also the conjunction of $$P$$ and $$\neg Q$$

Properties
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.

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

Grammatical
"p minus q."

"p without q."

Rhetorical
"p but not q."

"q is false, in spite of p."

Computer science
Bitwise operation: A&(~B)

Logical operation: A&&(!B)