Converse nonimplication

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In logic, converse nonimplication[1] is a logical connective which is the negation of the converse of implication.

Definition[edit]

which is the same as

Truth table[edit]

The truth table of .[2]

p q
T T F
T F F
F T T
F F F

Venn diagram[edit]

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.

Venn0010.svg

Properties[edit]

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[edit]

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[edit]

Grammatical[edit]

Rhetorical[edit]

"not A but B"

Colloquial[edit]

Boolean algebra[edit]

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.

1 0
x 0 1
and
y
1 1 1
0 0 1
0 1 x
and
y
1 0 1
0 0 0
0 1 x
then means
y
1 0 0
0 0 1
0 1 x
(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.

6 3 2 1
x 1 2 3 6
and
y
6 6 6 6 6
3 3 6 3 6
2 2 2 6 6
1 1 2 3 6
1 2 3 6 x
and
y
6 1 2 3 6
3 1 1 3 3
2 1 2 1 2
1 1 1 1 1
1 2 3 6 x
then means
y
6 1 1 1 1
3 1 2 1 2
2 1 1 3 3
1 1 2 3 6
1 2 3 6 x
(Codivisor 6) (Least Common Multiple) (Greatest Common Divisor) (x's greatest Divisor coprime with y)

Properties[edit]

Non-associative[edit]

iff #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 iff .

Non-commutative[edit]

  • iff #s6. Hence Converse Nonimplication is noncommutative.

Neutral and absorbing elements[edit]

  • 0 is a left neutral element () and a right absorbing element ().
  • , , and .
  • Implication is the dual of Converse Nonimplication #s7.

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
Step Make use of Resulting in
Definition
- .'s dual is +
- Involution complement
- De Morgan's laws applied once
- Commutative law

Computer science[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]

References[edit]