Residual operators are used when "material implication" is involved.
right residual : a (r\s) b ⇔ ∀x:x r a→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.
shows how you can type boolean (and other) operators in your Ampersand script.
Would you like a different explanation of the residual operators? explains them in natural language. for visualized examples about residual 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 ⇔ 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.
shows how you can type boolean (and other) operators in your Ampersand script.
Would you like a different explanation of the boolean operators? explains the boolean operators in terms of set theory.
To say things such as "the name of the owner", we want to string together multiple relations (viz. name and owner). Relational operators allow us to make such statements.
A relation can be altered by swapping the elements of every pair in the relation. Mathematically, is a different from . This operation is called the converse operator. It produces a new relation from an existing one. It is denoted by writing (pronounced 'wok' or ’flip’) after the relation name. This is how converse is defined:
If has type, then has type .
The composition operator is denoted by a semicolon between two terms. It is pronounced as 'composed with'. Let us take a look at composed with . Let and be two relations, with the target of r being the same as the source of s. Then the composition of and is defined by:
If has typeand has type, then has type .
shows how you can type boolean (and other) operators in your Ampersand script.
Would you like a different explanation of the relational operators? explains the relational operators in terms of set theory. explains them in natural language. for some algebraic rules about relational operators.
When a relation is used in a term, it stands for all pairs it contains at the moment it is evaluated. Those pairs (also referred to as the contents or population of the relation) can change over time as users add or delete pairs from it.
When a relation is used in a term, we can just use its name if that is unambiguous. For instance the name owner refers to RELATION owner[Person*Building] if that is the only relation the ampersand-compiler can link it to. In some cases, however the name alone is ambiguous. For example if there are two relations with the same name and different signatures. In such cases Ampersand will try to infer the type from the context. That however does not always succeed. In such cases, Ampersand generates an error message that asks you to remove the ambiguity by adding the correct type.
If a pair is an element of a relation , we write . Alternatively we may write , since we know that is a set.
For every concept , the term exists. It refers to the identity relation. It means that for every and we have:
The type of is . In Ampersand code you write I[C].
For every pair of concepts and the term refers to the complete relation. For every and we have:
The type of is . In Ampersand code you write V[A*B].
Would you like a different explanation of the primitive terms? explains the primitive terms in terms of set theory. for the explanation of primitive terms in natural language.