# 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 $$(a, b)$$ 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:

$$
r\smallsmile\ =\ { (b, a) | (a, b)∈r }
$$

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', in this case: $$r$$ composed with $$s$$.

The composition operation is defined as follows: 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$$, is a relation with signature $$(r;s)\_{\[A\times C]}\ =\ { (a, c) | ∃ b∈B\ ∙\ a\ r\ b ∧ b\ s\ c }$$

## Other explanation

Would you like a different explanation of the relational operators? [This page](https://ampersandtarski.gitbook.io/documentation/the-language-ampersand/terms/semantics-in-logic/relational-operators) explains the relational operators in logic. [This page](https://ampersandtarski.gitbook.io/documentation/the-language-ampersand/terms/semantics-in-natural-language/relational-operators-in-natural-language) explains them in natural language. [Click here](https://ampersandtarski.gitbook.io/documentation/the-language-ampersand/terms/semantics-in-algebra/relational-operators-in-algebra) for some algebraic rules about relational operators.