Material nonimplication

Material nonimplication or abjunction ( ) 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 .

Material nonimplication or abjunction ( ) 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 . ==Definition== ===Truth table=== ===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 219B 16 (8603 decimal): ↛. ==Natural language== ===Grammatical=== "p minus q." "p without q." ===Rhetorical=== "p but not q." "q is false, in spite of p." ==Computer science== Bitwise operation: P & ~Q. This is usually called "bit clear" (BIC) or "and not" (ANDN). Logical operation: P && !Q. ==See also== Implication Set difference ==References== ==External links==

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