A relation is by definition a subset of the Cartesian Product of the source and target sets. So, if two different relations r and s are defined on the same source A and target B, then the ordinary set operators can be applied to produce a new relation.
intersection : r∩s is the set that contains the elements that are contained in relation r as well as in s, or r∩s = {(x,y) ∣ (x,y)∈r∧(x,y)∈s}
union : r∪s is the set that contains all elements that are contained either in relation or in , or
difference : is the set that contains the elements of relation that are not contained in , or
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 the identical population yet a different type may have different complements.
shows how you can write these things in your Ampersand script.
Would you like a different explanation of the boolean operators? explains the boolean operators in logic.