Definition 1. Let $X$ and $Y$ be two sets. $X$ is a subset of $Y$, written $X \subseteq Y$, if
\[\forall a :\enspace a \in X \enspace\rimp\enspace a \in Y.\]$X$ is a proper subset of $Y$, written $X \subset Y$, if additionally $X \neq Y$.
Proposition 1. $\empt$ is the only set that is a subset of every non-empty set.
Proof
$\empt$ is a subset of every set since no $x \in \empt$. If $X$ is a subset of every non-empty set, then $X \subseteq \{X\}$. Were any $a \in X$, then $a = X$, then $X \in X$, contrary to irreflexivity of $\in$.
Proposition 2. Reflexivity. For any set $X$,
\[X \subseteq X.\]Proposition 3. Transitivity. For any sets $X$, $Y$ and $Z$,
\[X \subseteq Y \,\land\, Y\subseteq Z \enspace\rimp\enspace X \subseteq Z.\]Proposition 4. Antisymmetry. For any sets $X$ and $Y$,
\[X \subseteq Y \,\land\, Y \subseteq X \enspace\rimp\enspace X = Y.\]To summarize reflexivity, transitivity and antisymmetry of $\subseteq$:
Proposition 5. $\subseteq$ is a partial order.
References
- [1]
- Wikipedia contributors. Subset. Wikipedia, The Free Encyclopedia.
- [2]
- Thomas Jech. Set Theory. 3rd Millennium ed, rev. and expanded. Springer, 2002. p. 9.