Set Difference is Subset/Proof 1

Theorem

$S \setminus T \subseteq S$


Proof

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

The result follows from the definition of subset.

$\blacksquare$


Sources

  • 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets
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