Converse of a non-binary relation.

Question

In case of a binary relation $\rho$ between two sets A and B, $$\rho=\{(a,b) \mid a\in A \wedge b\in B\} \quad \&\quad \rho\subseteq A\times B $$ we define the converse as $$\rho ^{-1}=\{ (b,a) \mid (a,b)\in \rho \}$$ But in case of a finitary $n$-ary (for any arbitrary $n$) relation $\psi$ between $n$ sets $A_1,A_2, \ldots ,A_n$, $$\psi =\{ (a_1,a_2,\ldots ,a_n)\,|\,a_1\in A_1 \wedge a_2\in A_2 \wedge \ldots \wedge a_n\in A_n\}\quad \& \quad \psi\subseteq A_1\times A_2\times\ldots\times A_n$$ How to define $\psi ^{-1}$?

Answer 1

Just extend the definition of a converse for a binary relation.

Let $A_1, A_2, ..., A_n$ be sets, and let $\psi$ be a relation on $A_1, A_2, ..., A_n$.

Then the converse of $\psi$ would be:

$$\psi ^{-1} = \{(a_n,a_{n-1},...,a_1) \mid (a_1,a_2,...,a_n) \in \psi \}$$