Set Difference and Intersection form Partition/Corollary 1
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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.
$\blacksquare$