 (If \( \alpha, \beta \neq \emptyset \) , otherwise \( \alpha + \emptyset = \alpha, \emptyset + \beta = \beta \) ). This qualification is hidden in PM by the use of expressions for functional relations that are sometimes undefined.  The arithmetic sum of \(\alpha\) and \(\beta\) is the union of \(\alpha\) and \(\beta\) after they are made disjoint by summing the pairing the elements of \(\beta\) with elements of \(\{ \alpha \}\) and the elements of \(\alpha\) with the elements of \(\{ \beta \}\). (The Cartesian product of \( \gamma \) and \( \delta \), \( \gamma \times \delta \) is \( \{ \langle x, y \rangle | x \in \gamma \; \amp \; y \in \delta \} \)). The The classes \(\alpha\) and \(\beta\) are intersected with the empty class, \(\emptyset\), to adjust the type of the elements of the sum. It is more recognizable to contemporary set theory by the equivalent definition (subject to the same exception when   : 