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Semantics in sets

Primitive terms in set theory

Relations

When a relation is used in a term, it stands for the set of pairs it contains at the moment it is evaluated. That set (also referred to as the contents 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 simply 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 $(a,b)\in r$ . Alternatively we may write $a\ r\ b$ .

Identity

For every concept $C$ , the term $I_{[C]}$ represents the identity relation. It is defined by:

I_{[C]}\ =\ \{(c,c) |\ c\in C\}

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 $V_{[A*B]}$ represents the complete relation. It is defined by:

V_{[A*B]}\ =\ \{(a,b) |\ a\in A\ \wedge\ b\in B\}

Other explanation

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

Boolean operators in set theory

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 $r$ or in $s$ , or $r ∪ s\ =\ \{ (x,y)\ |\ (x,y) ∈ r ∨ (x,y) ∈ s \}$
difference : $r - s$ is the set that contains the elements of relation $r$ that are not contained in $s$ , or $r - s\ =\ \{ (x,y)\ |\ (x,y) ∈ r ∧ (x,y) ∉ s \}$

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 the identical population yet a different type may have different complements.

How to type boolean operators in your script

This page shows how you can write these things in your Ampersand script.

Other explanation

Would you like a different explanation of the boolean operators? This page explains the boolean operators in logic.

Relational operators in set theory

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 that contains pairs of the form 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 .

Composition

The composition operator is denoted by a semicolon ; between two terms. It is pronounced as 'composed with', in this case: composed with .

The composition operation is defined as follows: Let and be two relations, with the target of r being the same as the source of s. Then the composition of and , is a relation with signature

Other explanation

Would you like a different explanation of the relational operators? explains the relational operators in logic. explains them in natural language. for some algebraic rules about relational operators.

Boolean operators in set theory

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 $r$ or in $s$ , or $r ∪ s\ =\ \{ (x,y)\ |\ (x,y) ∈ r ∨ (x,y) ∈ s \}$
difference : $r - s$ is the set that contains the elements of relation $r$ that are not contained in $s$ , or $r - s\ =\ \{ (x,y)\ |\ (x,y) ∈ r ∧ (x,y) ∉ s \}$

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 the identical population yet a different type may have different complements.

How to type boolean operators in your script

This page shows how you can write these things in your Ampersand script.

Other explanation

Would you like a different explanation of the boolean operators? This page explains the boolean operators in logic.

Primitive terms in set theory

Relations

If a pair $(a,b)$ is an element of a relation $r$ , we write $(a,b)\in r$ . Alternatively we may write $a\ r\ b$ .

Identity

For every concept $C$ , the term $I_{[C]}$ represents the identity relation. It is defined by:

I_{[C]}\ =\ \{(c,c) |\ c\in C\}

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 $V_{[A*B]}$ represents the complete relation. It is defined by:

V_{[A*B]}\ =\ \{(a,b) |\ a\in A\ \wedge\ b\in B\}

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 logic. Click here for the explanation of primitive terms in natural language.

Relational operators in set theory

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

If has type , then has type .

Composition

The composition operator is denoted by a semicolon ; between two terms. It is pronounced as 'composed with', in this case: composed with .

The composition operation is defined as follows: Let and be two relations, with the target of r being the same as the source of s. Then the composition of and , is a relation with signature

Other explanation

Would you like a different explanation of the relational operators? explains the relational operators in logic. explains them in natural language. for some algebraic rules about relational operators.