Set Difference of Cartesian Products

Theorem

 * $\paren {S_1 \times S_2} \setminus \paren {T_1 \times T_2} = \paren {S_1 \times \paren {S_2 \setminus T_2} } \cup \paren {\paren {S_1 \setminus T_1} \times S_2}$

Proof
Let $\tuple {x, y} \in \paren {S_1 \times S_2} \setminus \paren {T_1 \times T_2}$.

Then:

The result follows from the definition of subset and set equality.