Set Difference is Subset

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Theorem

Set difference is a subset of the first set:

$S \setminus T \subseteq S$


Proof 1

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

The result follows from the definition of subset.

$\blacksquare$


Proof 2

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

$\blacksquare$


Sources