Cartesian Product

Tags

logic

For two sets \( A \) and \( B \), their cartesian product is the set of all ordered pairs \( (a, b) \) where \( a \in A \) and \( b \in B \). \[ A \times B = \{ (a, b) : a \in A \land b \in B \} \]

Inversion

We define an [see page 9, inversion] of a Cartesian product as the reversal of the order of the tuples of the product. So for example given \( R \subseteq A \times B \) an inversion \( R^{-1} \subseteq B \times A \) is \( \{ (b, a) : (a, b) \in R \} \).