Set is Subset of Itself
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Theorem
Every set is a subset of itself:
- $\forall S: S \subseteq S$
Thus, by definition, the relation is a subset of is reflexive.
Proof
\(\ds \forall x: \, \) | \(\ds \leftparen {x \in S}\) | \(\implies\) | \(\ds \rightparen {x \in S}\) | Law of Identity: | \(\quad\) a statement implies itself | |||||||||
\(\ds \leadsto \ \ \) | \(\ds S\) | \(\subseteq\) | \(\ds S\) | Definition of Subset |
$\blacksquare$
Sources
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