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 : $a\ (s/r)\ b\ \Leftrightarrow\ \forall x: b\ r\ x\rightarrow a\ s\ x$ . In words: $(a,b)$ is in the left residual of $s$ and $r$

  means that for every $x$ pair$(b,x)$ is in relation $r$ implies that pair $(a,x)$ is in $s$.

• diamond: $a (r♢s) b\ \Leftrightarrow\ \forall x: a\ r\ x\ =\ x\ s\ b$. In words: For every $x$, both $a\ r\ x$ and $x\ s\ b$ are true or both are false.

How to type boolean operators in your script

This page 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? This page explains them in natural language. Click here for visualized examples about residual operators.