Set Difference Intersection with First Set is Set Difference

From ProofWiki
Jump to navigation Jump to search


The intersection of the set difference with the first set is the set difference.

Let $S, T$ be sets.


$\left({S \setminus T}\right) \cap S = S \setminus T$

Proof 1

\(\displaystyle \left({S \setminus T}\right)\) \(\subseteq\) \(\displaystyle S\) Set Difference is Subset
\(\displaystyle \implies \ \ \) \(\displaystyle \left({S \setminus T}\right) \cap S\) \(=\) \(\displaystyle S \setminus T\) Intersection with Subset is Subset‎


Proof 2

\(\displaystyle \left({S \setminus T}\right) \cap S\) \(=\) \(\displaystyle \left({S \cap S}\right) \setminus T\) Intersection with Set Difference is Set Difference with Intersection
\(\displaystyle \) \(=\) \(\displaystyle S \setminus T\) Intersection is Idempotent