Cartesian Product with Complement

Theorem
Let $S$ and $T$ be sets.

Let $A \subseteq S$ and $B \subseteq T$ be subsets of $S$ and $T$, respectively.

Denote with $\complement_S \left({A}\right)$ the relative complement of $A$ in $S$.

Then:

Proof
By definition of relative complement we have:


 * $\complement_S \left({A}\right) = S \setminus A$

where $S \setminus A$ denotes set difference.

By Cartesian Product Distributes over Set Difference, we have:


 * $\left({S \setminus A}\right) \times T = \left({S \times T}\right) \setminus \left({A \times T}\right)$

and the latter equals $\complement_{S \times T} \left({A \times T}\right)$.

In conclusion, we obtain:


 * $\complement_S \left({A}\right) \times T = \complement_{S \times T} \left({A \times T}\right)$

as desired.

Mutatis mutandis, the other statement follows from this argument as well.