Converse nonimplication

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.

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

${\displaystyle \scriptstyle {x^{c}}\!}$ 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 ${\displaystyle \scriptstyle {y_{\vee }x}\!}$ 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 ${\displaystyle \scriptstyle {y_{\wedge }x}}$ 1 2 3 6 x	then ${\displaystyle \scriptstyle {y\nleftarrow x}\!}$ means	y 6 1 1 1 1 3 1 2 1 2 2 1 1 3 3 1 1 2 3 6 ${\displaystyle \scriptstyle {y\nleftarrow x}\!}$ 1 2 3 6 x
(Co-divisor 6)		(Least common multiple)		(Greatest common divisor)		(x's greatest divisor coprime with y)

${\displaystyle r\nleftarrow (q\nleftarrow p)=(r\nleftarrow q)\nleftarrow p}$ if and only if ${\displaystyle rp=0}$ #s5 (In a two-element Boolean algebra the latter condition is reduced to ${\displaystyle r=0}$ or ${\displaystyle p=0}$). Hence in a nontrivial Boolean algebra Converse Nonimplication is nonassociative. $${\displaystyle {\begin{aligned}(r\nleftarrow q)\nleftarrow p&=r'q\nleftarrow p&{\text{(by definition)}}\\&=(r'q)'p&{\text{(by definition)}}\\&=(r+q')p&{\text{(De Morgan's laws)}}\\&=(r+r'q')p&{\text{(Absorption law)}}\\&=rp+r'q'p\\&=rp+r'(q\nleftarrow p)&{\text{(by definition)}}\\&=rp+r\nleftarrow (q\nleftarrow p)&{\text{(by definition)}}\\\end{aligned}}}$$

Clearly, it is associative if and only if ${\displaystyle rp=0}$.

• ${\displaystyle q\nleftarrow p=p\nleftarrow q}$ if and only if ${\displaystyle q=p}$ #s6. Hence Converse Nonimplication is noncommutative.

• 0 is a left neutral element (${\displaystyle 0\nleftarrow p=p}$) and a right absorbing element (${\displaystyle {p\nleftarrow 0=0}}$).
• ${\displaystyle 1\nleftarrow p=0}$, ${\displaystyle p\nleftarrow 1=p'}$, and ${\displaystyle p\nleftarrow p=0}$.
• Implication ${\displaystyle q\rightarrow p}$ is the dual of converse nonimplication ${\displaystyle q\nleftarrow p}$ #s7.