 Venn diagram 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 $\neg (P\leftarrow Q)$ ### Truth table

The truth table of $P\nleftarrow Q$ .

 $P$ $Q$ $P\nleftarrow Q$ True True False True False False False True True False False False

## Notation

Converse nonimplication is notated ${\textstyle p\nleftarrow q}$ , which is the left arrow from converse implication (${\textstyle \leftarrow }$ ), negated with a stroke (/).

Alternatives include

• ${\textstyle p\not \subset q}$ , which combines converse implication's $\subset$ , negated with a stroke (/).
• ${\textstyle p{\tilde {\leftarrow ))q}$ , which combines converse implication's left arrow (${\textstyle \leftarrow }$ ) with negation's tilde (${\textstyle \sim }$ ).
• 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

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 ${\textstyle q\nleftarrow p=q'p}$ .

Example of a 2-element Boolean algebra: the 2 elements {0,1} with 0 as zero and 1 as unity element, operators ${\textstyle \sim }$ as complement operator, ${\textstyle \vee }$ as join operator and ${\textstyle \wedge }$ as meet operator, build the Boolean algebra of propositional logic.

 0 1 ${\textstyle {}\sim x}$ 1 0 x
and
1 0 y 1 1 0 1 ${\textstyle y_{\vee }x}$ x
and
1 0 y 0 1 0 0 ${\textstyle y_{\wedge }x}$ x
then ${y\nleftarrow x}\!$ means
1 0 y 0 0 0 1 ${y\nleftarrow x}\!$ 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 ${^{c))\!$ (codivisor of 6) as complement operator, ${_{\vee ))\!$ (least common multiple) as join operator and ${_{\wedge ))\!$ (greatest common divisor) as meet operator, build a Boolean algebra.

 1 2 3 6 ${x^{c))\!$ 6 3 2 1 x
and
6 3 2 1 y 6 6 6 6 3 6 3 6 2 2 6 6 1 2 3 6 ${y_{\vee }x}\!$ x
and
6 3 2 1 y 1 2 3 6 1 1 3 3 1 2 1 2 1 1 1 1 ${y_{\wedge }x)$ x
then ${y\nleftarrow x}\!$ means
6 3 2 1 y 1 1 1 1 1 2 1 2 1 1 3 3 1 2 3 6 ${y\nleftarrow x}\!$ x
(Codivisor 6) (Least common multiple) (Greatest common divisor) (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{aligned}(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{aligned)) 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

• 0 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.

Converse Nonimplication is noncommutative
Step Make use of Resulting in
s.1 Definition ${q{\tilde {\leftarrow ))p=q'p\,}\!$ s.2 Definition ${p{\tilde {\leftarrow ))q=p'q\,}\!$ s.3 s.1 s.2 ${q{\tilde {\leftarrow ))p=p{\tilde {\leftarrow ))q\ \Leftrightarrow \ q'p=qp'\,}\!$ s.4 ${q\,}\!$ ${=\,}\!$ ${q.1\,}\!$ s.5 s.4.right - expand Unit element ${=\,}\!$ ${q.(p+p')\,}\!$ s.6 s.5.right - evaluate expression ${=\,}\!$ ${qp+qp'\,}\!$ s.7 s.4.left = s.6.right ${q=qp+qp'\,}\!$ s.8 ${q'p=qp'\,}\!$ ${\Rightarrow \,}\!$ ${qp+qp'=qp+q'p\,}\!$ s.9 s.8 - regroup common factors ${\Rightarrow \,}\!$ ${q.(p+p')=(q+q').p\,}\!$ s.10 s.9 - join of complements equals unity ${\Rightarrow \,}\!$ ${q.1=1.p\,}\!$ s.11 s.10.right - evaluate expression ${\Rightarrow \,}\!$ ${q=p\,}\!$ s.12 s.8 s.11 ${q'p=qp'\ \Rightarrow \ q=p\,}\!$ s.13 ${q=p\ \Rightarrow \ q'p=qp'\,}\!$ s.14 s.12 s.13 ${q=p\ \Leftrightarrow \ q'p=qp'\,}\!$ s.15 s.3 s.14 ${q{\tilde {\leftarrow ))p=p{\tilde {\leftarrow ))q\ \Leftrightarrow \ q=p\,}\!$ Implication is the dual of Converse Nonimplication
Step Make use of Resulting in
s.1 Definition ${\operatorname {dual} (q{\tilde {\leftarrow ))p)\,}\!$ ${=\,}\!$ ${\operatorname {dual} (q'p)\,}\!$ s.2 s.1.right - .'s dual is + ${=\,}\!$ ${q'+p\,}\!$ s.3 s.2.right - Involution complement ${=\,}\!$ ${(q'+p)''\,}\!$ s.4 s.3.right - De Morgan's laws applied once ${=\,}\!$ ${(qp')'\,}\!$ s.5 s.4.right - Commutative law ${=\,}\!$ ${(p'q)'\,}\!$ s.6 s.5.right ${=\,}\!$ ${(p{\tilde {\leftarrow ))q)'\,}\!$ s.7 s.6.right ${=\,}\!$ ${p\leftarrow q\,}\!$ s.8 s.7.right ${=\,}\!$ ${q\rightarrow p\,}\!$ s.9 s.1.left = s.8.right ${\operatorname {dual} (q{\tilde {\leftarrow ))p)=q\rightarrow p\,}\!$ ## 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.

1. ^ Lehtonen, Eero, and Poikonen, J.H.
2. ^ Knuth 2011, p. 49
3. ^ "A Visual Explanation of SQL Joins". 11 October 2007.
• Media related to Converse nonimplication at Wikimedia Commons