Set Difference and Intersection form Partition/Corollary 1

Corollary to Set Difference and Intersection form Partition

Let $S$ and $T$ be sets such that:

$S \setminus T \ne \O$

$T \setminus S \ne \O$

$S \cap T \ne \O$

Then $S \setminus T$, $T \setminus S$ and $S \cap T$ form a partition of $S \cup T$, the union of $S$ and $T$.

Proof

From Set Difference and Intersection form Partition:

$S \setminus T$ and $S \cap T$ form a partition of $S$

$T \setminus S$ and $S \cap T$ form a partition of $T$

From Set Difference is Disjoint with Reverse:

$\paren {S \setminus T} \cap \paren {T \setminus S} = \O$

So:

$S \cup T = \paren {S \setminus T} \cup \paren {S \cap T} \cup \paren {T \setminus S} \cup \paren {S \cap T}$

and the result follows.

$\blacksquare$