Cartesian Product with Complement

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Theorem

Let $S$ and $T$ be sets.

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

Let $\relcomp S A$ denote the relative complement of $A$ in $S$.


Then:

\(\displaystyle \relcomp S A \times T\) \(=\) \(\displaystyle \relcomp {S \times T} {A \times T}\)
\(\displaystyle S \times \relcomp T B\) \(=\) \(\displaystyle \relcomp {S \times T} {S \times B}\)


Proof

By definition of relative complement we have:

$\relcomp S A = S \setminus A$

where $S \setminus A$ denotes set difference.

By Cartesian Product Distributes over Set Difference, we have:

$\paren {S \setminus A} \times T = \paren {S \times T} \setminus \paren {A \times T}$

and the latter equals $\relcomp {S \times T} {A \times T}$.

In conclusion, we obtain:

$\relcomp S A \times T = \relcomp {S \times T} {A \times T}$

as desired.


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

$\blacksquare$