Converse nonimplication: Difference between revisions
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===Properties=== |
===Properties=== |
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====Non-associative==== |
====Non-associative==== |
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<math>\scriptstyle{r \nleftarrow (q \nleftarrow p)=(r \nleftarrow q) \nleftarrow p}</math> iff <math>\scriptstyle{rp=0}</math> |
<math>\scriptstyle{r \nleftarrow (q \nleftarrow p)=(r \nleftarrow q) \nleftarrow p}</math> iff <math>\scriptstyle{rp=0}</math> <ref name="NonAssociative"/> (In a [[two-element Boolean algebra]] the latter condition is reduced to <math>\scriptstyle{r=0}</math> or <math>\scriptstyle{p=0}</math>). Hence in a nontrivial Boolean algebra Converse Nonimplication is '''nonassociative'''. |
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::<math> |
::<math> |
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====Non-commutative==== |
====Non-commutative==== |
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* <math>\scriptstyle{q \nleftarrow p=p \nleftarrow q\,}\!</math> iff <math>\scriptstyle{q=p\,}\!</math> |
* <math>\scriptstyle{q \nleftarrow p=p \nleftarrow q\,}\!</math> iff <math>\scriptstyle{q=p\,}\!</math> <ref name="NonCommutative"/>. Hence Converse Nonimplication is '''noncommutative'''. |
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====Neutral and absorbing elements==== |
====Neutral and absorbing elements==== |
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* {{math|size=100%|0}} is a left [[neutral element]] (<math>\scriptstyle{0 \nleftarrow p=p}\!</math>) and a right [[absorbing element]] (<math>\scriptstyle{p \nleftarrow 0=0}\!</math>). |
* {{math|size=100%|0}} is a left [[neutral element]] (<math>\scriptstyle{0 \nleftarrow p=p}\!</math>) and a right [[absorbing element]] (<math>\scriptstyle{p \nleftarrow 0=0}\!</math>). |
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* <math>\scriptstyle{1 \nleftarrow p=0}\!</math>, <math>\scriptstyle{p \nleftarrow 1=p'}\!</math>, and <math>\scriptstyle{p \nleftarrow p=0}\!</math>. |
* <math>\scriptstyle{1 \nleftarrow p=0}\!</math>, <math>\scriptstyle{p \nleftarrow 1=p'}\!</math>, and <math>\scriptstyle{p \nleftarrow p=0}\!</math>. |
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* Implication <math>\scriptstyle{q \rightarrow p}\!</math> is the dual of Converse Nonimplication <math>\scriptstyle{q \nleftarrow p}\!</math> |
* Implication <math>\scriptstyle{q \rightarrow p}\!</math> is the dual of Converse Nonimplication <math>\scriptstyle{q \nleftarrow p}\!</math><ref name ="Dual" />. |
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Revision as of 07:50, 10 October 2016
In logic, converse nonimplication[1] is a logical connective which is the negation of the converse of implication.
Definition
which is the same as
Truth table
The truth table of .[2]
p | q | ⊄ |
---|---|---|
T | T | F |
T | F | F |
F | T | T |
F | F | F |
Venn diagram
The Venn Diagram of "It is not the case that B implies A" (the red area is true).
Also related to the relative complement (set theory), where the relative complement of A in B is denoted B ∖ A.
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
Symbol
Alternatives for are
- : combines Converse implication's left arrow() with Negation's tilde().
- : uses prefixed capital letter.
- : combines Converse implication's left arrow() denied by means of a stroke(/).
Natural language
Grammatical
This section is empty. You can help by adding to it. (February 2011) |
Rhetorical
"not A but B"
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 propositional logic.
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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 (codivisor 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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(Codivisor 6) | (Least Common Multiple) | (Greatest Common Divisor) | (x's greatest Divisor coprime with y) |
Properties
Non-associative
iff [3] (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 iff .
Non-commutative
- iff [4]. 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 [5].
Converse Nonimplication is noncommutative | ||||
---|---|---|---|---|
Step | Make use of | Resulting in | ||
Definition | ||||
Definition | ||||
- expand Unit element | ||||
- evaluate expression | ||||
- regroup common factors | ||||
- join of complements equals unity | ||||
- evaluate expression | ||||
Implication is the dual of Converse Nonimplication | ||||
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Step | Make use of | Resulting in | ||
Definition | ||||
- .'s dual is + | ||||
- Involution complement | ||||
- De Morgan's laws applied once | ||||
- Commutative law | ||||
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.[6]
References
- ^ Lehtonen, Eero, and Poikonen, J.H.
- ^ Knuth 2011, p. 49
- ^ Cite error: The named reference
NonAssociative
was invoked but never defined (see the help page). - ^ Cite error: The named reference
NonCommutative
was invoked but never defined (see the help page). - ^ Cite error: The named reference
Dual
was invoked but never defined (see the help page). - ^ http://www.codinghorror.com/blog/2007/10/a-visual-explanation-of-sql-joins.html
- Knuth, Donald E. (2011). The Art of Computer Programming, Volume 4A: Combinatorial Algorithms, Part 1 (1st ed.). Addison-Wesley Professional. ISBN 0-201-03804-8.
{{cite book}}
: Invalid|ref=harv
(help)