Converse nonimplication
In
Definition
Converse nonimplication is notated , or , and is logically equivalent to and .
Truth table
The truth table of .[2]
F | F | F |
F | T | T |
T | F | F |
T | T | F |
Notation
Converse nonimplication is notated , which is the left arrow from
Alternatives include
- , which combines converse implication's, negated with a stroke (/).
- , which combines converse implication'sleft arrow () with negation's tilde ().
- 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
This section is empty. You can help by adding to it. (February 2011) |
Boolean algebra
Converse Nonimplication in a general Boolean algebra is defined as .
Example of a 2-element Boolean algebra: the 2 elements {0,1} with 0 as zero and 1 as unity element, operators as complement operator, as join operator and as meet operator, build the Boolean algebra of
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and |
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and |
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then means |
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(Negation) | (Inclusive or) | (And) | (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 (co-divisor of 6) as complement operator, (least common multiple) as join operator and (greatest common divisor) as meet operator, build a Boolean algebra.
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and |
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and |
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then means |
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(Co-divisor 6) | (Least common multiple) | (Greatest common divisor) | (x's greatest divisor coprime with y)
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Properties
Non-associative
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 .
Non-commutative
- if and only if #s6. Hence Converse Nonimplication is noncommutative.
Neutral and absorbing elements
- 0 is a left neutral element() and a right absorbing element ().
- , , and .
- Implication is the dual of converse nonimplication #s7.
Converse Nonimplication is noncommutative | ||||
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Step | Make use of | Resulting in | ||
s.1 | Definition | |||
s.2 | Definition | |||
s.3 | s.1 s.2 | |||
s.4 | ||||
s.5 | s.4.right - expand Unit element | |||
s.6 | s.5.right - evaluate expression | |||
s.7 | s.4.left = s.6.right | |||
s.8 | ||||
s.9 | s.8 - regroup common factors | |||
s.10 | s.9 - join of complements equals unity | |||
s.11 | s.10.right - evaluate expression | |||
s.12 | s.8 s.11 | |||
s.13 | ||||
s.14 | s.12 s.13 | |||
s.15 | s.3 s.14 |
Implication is the dual of Converse Nonimplication | ||||
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Step | Make use of | Resulting in | ||
s.1 | Definition | |||
s.2 | s.1.right - .'s dual is + | |||
s.3 | s.2.right - Involution complement | |||
s.4 | s.3.right - De Morgan's laws applied once | |||
s.5 | s.4.right - Commutative law
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s.6 | s.5.right | |||
s.7 | s.6.right | |||
s.8 | s.7.right | |||
s.9 | s.1.left = s.8.right |
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.[3]
References
- ^ 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.
- ISBN 978-0-201-03804-0.
External links
- Media related to Converse nonimplication at Wikimedia Commons