Boolean operators

The notation $a\ r\ b$ means that the pair (a,b) is in relation $r$. This page defines when pair (a,b) is in relation $r ∩ s$ (the intersection of $r$ and $s$), $r ∪ s$ (the union of $r$ and $s$), $r-s$ (the difference of $r$ and $s$).

• intersection : $a\ (r ∩ s)\ b\ \Leftrightarrow\ a\ r\ b\ ∧\ a\ s\ b$ . In other words: if the pair $(a,b)$ is both in relation $r$ and $s$, then it is in the intersection of $r$ and $s$.

• union : $a\ (r ∪ s)\ b\ \Leftrightarrow\ a\ r\ b\ \vee\ a\ s\ b$ . In words: if the pair $(a,b)$ is in the relation $r$ or in $s$, then it is in the union of $r$ and $s$.

• difference : $a\ (r-s)\ b\ \Leftrightarrow\ a\ r\ b\ ∧\ \neg(a\ s\ b)$. In other words, the term $r-s$ contains all pairs from $r$ that are not in $s$.

The complement (or negation) of a relation $r_{[A x B]}$ is defined by means of the difference operator:

• complement : If $r$ is defined as $r_{A\times B}$, then $\overline{r}$ is the set of all tuples in $A\times B$ (the Cartesian product) that are not contained in $r$. So $\overline{r} = V_{[A\times B]} - r$

Note that the complement is defined in terms of $A$ and $B$. So, two relations with an identical population yet a different type may have different complements.

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 boolean operators? This page explains the boolean operators in terms of set theory.