Residual operators

Residual operators are used when "material implication" is involved.

• right residual : $a\ (r\backslash s)\ b\ \Leftrightarrow\ \forall x: x\ r\ a\rightarrow x\ s\ b$ . In other words: $(a,b)$ is in the right residual of $r$ and $s$ means that for every $x$, pair $(x,a)$ is in relation $r$ implies that pair $(x,b)$ is in $s$.

• left residual : . In words: is in the left residual of and

  means that for every pair is in relation implies that pair is in .

• diamond: . In words: For every , both and are true or both are false.

How to type boolean operators in your script

shows how you can type boolean (and other) operators in your Ampersand script.

Other explanation

Would you like a different explanation of the residual operators? explains them in natural language. for visualized examples about residual operators.