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 explains the relational operators in logic. This page explains them in natural language. Click here for some algebraic rules about relational operators.