Converse nonimplication

of P \nleftarrow Q (the red area is true) In logic, converse nonimplication is a logical connective which is the negation of converse implication (equivalently, the negation of the converse of implication).

of P \nleftarrow Q (the red area is true)]] In logic, converse nonimplication is a logical connective which is the negation of converse implication (equivalently, the negation of the converse of implication). ==Definition== Converse nonimplication is notated P \nleftarrow Q , or P \not \subset Q , and is logically equivalent to \neg (P \leftarrow Q) and \neg P \wedge Q . ===Truth table=== The truth table of A \nleftarrow B . ==Notation== Converse nonimplication is notated p \nleftarrow q , which is the left arrow from converse implication ( \leftarrow ), negated with a stroke ( ). Alternatives include p \not\subset q , which combines converse implication's \subset , negated with a stroke ( ). p \tilde{\leftarrow} q , which combines converse implication's left arrow ( \leftarrow ) with negation's tilde ( \sim ). Mpq, in Bocheński notation ==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 converse nonimplication ==Natural language== ===Grammatical=== 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). ===Rhetorical=== Q does not imply P. ===Colloquial=== Not P, but Q. ==Boolean algebra== Converse nonimplication in a general Boolean algebra is defined as 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 \sim as complement operator, \vee as join operator and \wedge as meet operator, build the Boolean algebra of propositional logic. | style="border:none;" |and |style="border:none;" | | style="border:none;" |and |style="border:none;" | | style="border:none;" |then \scriptstyle{y \nleftarrow x}\! means |style="border:none;" | |- | style="border:none;" |(Negation) | style="border:none;" | | style="border:none;" |(Inclusive or) | style="border:none;" | | style="border:none;" |(And) | style="border:none;" | | style="border:none;" |(Converse nonimplication) |} 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 \scriptstyle{ ^{c}}\! (co-divisor of 6) as complement operator, \scriptstyle{_\vee}\! (least common multiple) as join operator and \scriptstyle{_\wedge}\! (greatest common divisor) as meet operator, build a Boolean algebra. | style="border:none;" |and |style="border:none;" | | style="border:none;" |and |style="border:none;" | | style="border:none;" |then \scriptstyle{y \nleftarrow x}\! means |style="border:none;" | |- | style="border:none;" |(Co-divisor 6) | style="border:none;" | | style="border:none;" |(Least common multiple) | style="border:none;" | | style="border:none;" |(Greatest common divisor) | style="border:none;" | | style="border:none;" |(x's greatest divisor coprime with y) |} ===Properties=== ====Non-associative==== r \nleftarrow (q \nleftarrow p) = (r \nleftarrow q) \nleftarrow p if and only if rp = 0 #s5 (In a two-element Boolean algebra the latter condition is reduced to r = 0 or p=0 ). Hence in a nontrivial Boolean algebra converse nonimplication is nonassociative. \begin{align} (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{align} Clearly, it is associative if and only if rp=0 . ====Non-commutative==== q \nleftarrow p=p \nleftarrow q if and only if q = p #s6. Hence converse nonimplication is noncommutative. ====Neutral and absorbing elements==== is a left neutral element ( 0 \nleftarrow p=p ) and a right absorbing element ( {p \nleftarrow 0=0} ). 1 \nleftarrow p=0 , p \nleftarrow 1=p' , and p \nleftarrow p=0 . Implication q \rightarrow p is the dual of converse nonimplication q \nleftarrow p #s7. ==Computer science== 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. ==References== ==External links==

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