In logic, converse nonimplication[1] is a logical connective which is the negation of converse implication (equivalently, the negation of the converse of implication).
Converse nonimplication is notated P ↚ ↚ --> Q {\displaystyle P\nleftarrow Q} , or P ⊄ Q {\displaystyle P\not \subset Q} , and is logically equivalent to ¬ ¬ --> ( P ← ← --> Q ) {\displaystyle \neg (P\leftarrow Q)} and ¬ ¬ --> P ∧ ∧ --> Q {\displaystyle \neg P\wedge Q} .
The truth table of A ↚ ↚ --> B {\displaystyle A\nleftarrow B} .[2]
Converse nonimplication is notated p ↚ ↚ --> q {\textstyle p\nleftarrow q} , which is the left arrow from converse implication ( ← ← --> {\textstyle \leftarrow } ), negated with a stroke (/).
Alternatives include
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 converse nonimplication
Example,
If it rains (P) then I get wet (Q), just because I am wet (Q) does not mean it is raining, in reality I went to a pool party with the co-ed staff, in my clothes (~P) and that is why I am facilitating this lecture in this state (Q).
Q does not imply P.
Converse Nonimplication in a general Boolean algebra is defined as q ↚ ↚ --> p = q ′ p {\textstyle q\nleftarrow p=q'p} .
Example of a 2-element Boolean algebra: the 2 elements {0,1} with 0 as zero and 1 as unity element, operators ∼ ∼ --> {\textstyle \sim } as complement operator, ∨ ∨ --> {\textstyle \vee } as join operator and ∧ ∧ --> {\textstyle \wedge } as meet operator, build the Boolean algebra of propositional logic.
Example of a 4-element Boolean algebra: the 4 divisors {1,2,3,6} of 6 with 1 as zero and 6 as unity element, operators c {\displaystyle \scriptstyle {^{c}}\!} (co-divisor of 6) as complement operator, ∨ ∨ --> {\displaystyle \scriptstyle {_{\vee }}\!} (least common multiple) as join operator and ∧ ∧ --> {\displaystyle \scriptstyle {_{\wedge }}\!} (greatest common divisor) as meet operator, build a Boolean algebra.
r ↚ ↚ --> ( q ↚ ↚ --> p ) = ( r ↚ ↚ --> q ) ↚ ↚ --> p {\displaystyle r\nleftarrow (q\nleftarrow p)=(r\nleftarrow q)\nleftarrow p} if and only if r p = 0 {\displaystyle rp=0} #s5 (In a two-element Boolean algebra the latter condition is reduced to r = 0 {\displaystyle r=0} or p = 0 {\displaystyle p=0} ). Hence in a nontrivial Boolean algebra Converse Nonimplication is nonassociative. ( r ↚ ↚ --> q ) ↚ ↚ --> p = r ′ q ↚ ↚ --> p (by definition) = ( r ′ q ) ′ p (by definition) = ( r + q ′ ) p (De Morgan's laws) = ( r + r ′ q ′ ) p (Absorption law) = r p + r ′ q ′ p = r p + r ′ ( q ↚ ↚ --> p ) (by definition) = r p + r ↚ ↚ --> ( q ↚ ↚ --> p ) (by definition) {\displaystyle {\begin{aligned}(r\nleftarrow q)\nleftarrow p&=r'q\nleftarrow p&{\text{(by definition)}}\\&=(r'q)'p&{\text{(by definition)}}\\&=(r+q')p&{\text{(De Morgan's laws)}}\\&=(r+r'q')p&{\text{(Absorption law)}}\\&=rp+r'q'p\\&=rp+r'(q\nleftarrow p)&{\text{(by definition)}}\\&=rp+r\nleftarrow (q\nleftarrow p)&{\text{(by definition)}}\\\end{aligned}}}
Clearly, it is associative if and only if r p = 0 {\displaystyle rp=0} .
An example for converse nonimplication in computer science can be found when performing a right outer join on a set of tables from a database, if records not matching the join-condition from the "left" table are being excluded.[3]
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