Set Difference is Subset/Proof 1
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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