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:
\(\ds \relcomp S A \times T\) | \(=\) | \(\ds \relcomp {S \times T} {A \times T}\) | ||||||||||||
\(\ds S \times \relcomp T B\) | \(=\) | \(\ds \relcomp {S \times T} {S \times B}\) |
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.
$\blacksquare$