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 : . In words: if the pair is in the relation or in , then it is in the union of and .

• difference : . In other words, the term contains all pairs from that are not in .

The complement (or negation) of a relation is defined by means of the difference operator:

• complement : If is defined as , then is the set of all tuples in (the Cartesian product) that are not contained in . So

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

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