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.

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.

Relations

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 $(a,b)$ is an element of a relation $r$, we write $a\ r\ b$. Alternatively we may write $(a,b)\in r$ , since we know that $r$ is a set.

Identity

For every concept $C$, the term $I_{[C]}$ exists. It refers to the identity relation. It means that for every $a\in C$ and $b\in C$ we have:

The type of $I_{[C]}$ is $[C*C]$. In Ampersand code you write I[C].

Complete relation

For every pair of concepts $A$ and $B$ the term $V_{[A*B]}$ refers to the complete relation. For every $a\in A$ and $b\in B$ we have:

The type of $V_{[A*B]}$ is $[A*B]$. In Ampersand code you write V[A*B].

Other explanation

Would you like a different explanation of the primitive terms? This page explains the primitive terms in terms of set theory. Click here for the explanation of primitive terms in natural language.

Purpose of relational operators

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.

Converse

A relation can be altered by swapping the elements of every pair in the relation. Mathematically, $(a, b)$ is a different from $(b,a)$. This operation is called the converse operator. It produces a new relation from an existing one. It is denoted by writing \smallsmile\ (pronounced 'wok' or ’flip’) after the relation name. This is how converse is defined:

If $r$ has type$[A\times B]$, then r\smallsmile\ has type $[B\times A]$.

Composition

The composition operator is denoted by a semicolon $;$ between two terms. It is pronounced as 'composed with'. Let us take a look at $r$ composed with $s$. Let $r_{[A\times B]}$ and $s_{[B\times C]}$ be two relations, with the target of r being the same as the source of s. Then the composition of $r$ and $s$ is defined by:

If $r$ has type$[A\times B]$and $s$has type$[B\times C]$, then $r;s$ has type $[A\times C]$.

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 relational operators? This page explains the relational operators in terms of set theory. This page explains them in natural language. Click here for some algebraic rules about relational operators.