Set Difference Union Intersection/Proof 1
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Theorem
- $S = \paren {S \setminus T} \cup \paren {S \cap T}$
Proof
| \(\ds \paren {S \setminus T} \cup \paren {S \cap T}\) | \(=\) | \(\ds \paren {\paren {S \setminus T} \cup S} \cap \paren {\paren {S \setminus T} \cup T}\) | Union Distributes over Intersection | |||||||||||
| \(\ds \) | \(=\) | \(\ds S \cap \paren {\paren {S \setminus T} \cup T}\) | Set Difference Union First Set is First Set | |||||||||||
| \(\ds \) | \(=\) | \(\ds S \cap \paren {S \cup T}\) | Set Difference Union Second Set is Union | |||||||||||
| \(\ds \) | \(=\) | \(\ds S\) | Intersection Absorbs Union |
$\blacksquare$