Set Difference as Symmetric Difference with Intersection
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Theorem
- $S \setminus T = S \symdif \paren {S \cap T}$
where:
- $S \setminus T$ denotes set difference
- $S \symdif T$ denotes set symmetric difference
- $S \cap T$ denotes set intersection.
Proof
\(\ds S \symdif \paren {S \cap T}\) | \(=\) | \(\ds \paren {S \setminus \paren {S \cap T} } \cup \paren {\paren {S \cap T} \setminus S}\) | Definition of Symmetric Difference | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {S \setminus \paren {S \cap T} } \cup \O\) | Set Difference of Intersection with Set is Empty Set | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {S \setminus T} \cup \O\) | Set Difference with Intersection is Difference | |||||||||||
\(\ds \) | \(=\) | \(\ds S \setminus T\) | Union with Empty Set |
$\blacksquare$