Converse nonimplication
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(the red area is true)
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 , or , and is logically equivalent to and .
The truth table of .[2]
| F | F | F |
| F | T | T |
| T | F | F |
| T | T | F |
Converse nonimplication is notated , which is the left arrow from converse implication (), 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.
Not P, but Q.
Converse nonimplication in a general Boolean algebra is defined as .
if and only if #s5 (In a two-element Boolean algebra the latter condition is reduced to or ). Hence in a nontrivial Boolean algebra converse nonimplication is nonassociative.
Clearly, it is associative if and only if .
Neutral and absorbing elements
[edit]
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]
- ^ Lehtonen, Eero, and Poikonen, J.H.
- ^ Knuth 2011, p. 49
- ^ "A Visual Explanation of SQL Joins". 11 October 2007. Archived from the original on 15 February 2014. Retrieved 24 March 2013.
- Knuth, Donald E. (2011). The Art of Computer Programming, Volume 4A: Combinatorial Algorithms, Part 1 (1st ed.). Addison-Wesley Professional. ISBN 978-0-201-03804-0.
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