Set Difference is Subset

From ProofWiki
Jump to navigation Jump to search


Set difference is a subset of the first set:

$S \setminus T \subseteq S$

Proof 1

\(\ds x \in S \setminus T\) \(\leadsto\) \(\ds x \in S \land x \notin T\) Definition of Set Difference
\(\ds \) \(\leadsto\) \(\ds x \in S\) Rule of Simplification

The result follows from the definition of subset.


Proof 2

\(\ds S \setminus T\) \(=\) \(\ds S \cap \complement_S \left({T}\right)\) Set Difference as Intersection with Relative Complement
\(\ds \) \(\subseteq\) \(\ds S\) Intersection is Subset