Equal Set Differences iff Equal Intersections/Proof 1
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Theorem
- $R \setminus S = R \setminus T \iff R \cap S = R \cap T$
Proof
| \(\ds R \setminus S\) | \(=\) | \(\ds R \setminus T\) | ||||||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds \set {x \in R: x \notin S}\) | \(=\) | \(\ds \set {x \in R: x \notin T}\) | Definition of Set Difference | ||||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds \forall x \in R: \, \) | \(\ds x \notin S\) | \(\iff\) | \(\ds x \notin T\) | ||||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds \forall x \in R: \, \) | \(\ds x \in S\) | \(\iff\) | \(\ds x \in T\) | ||||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds \set {\paren {x \in R} \land \paren {x \in S} }\) | \(=\) | \(\ds \set {\paren {x \in R} \land \paren {x \in T} }\) | |||||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds R \cap S\) | \(=\) | \(\ds R \cap T\) | Definition of Set Intersection |
$\blacksquare$