Element Inclusion vs. Subsets

Examples: Let $A = \{a, b, c, d\}$, $B = \{a, b, d\}$, and $C = \{a, b, z\}$. We see that $B \subset A$, while $C \not\subset A$. (Is $A$ a subset of $B$? Why or why not?)

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Now that we know what subsets are, we can officially define what set equality is: if $A$ and $B$ are sets such that $A \subseteq B$ and $B \subseteq A$, it must hold that $A = B$.

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One additional special set that we should know about is the empty set, written either as $\emptyset$ (

$\emptyset$

) or as $\{\}$ (

$\{\}$

).

Fun facts: If a set $S$ doesn't contain $\emptyset$ as one of its elements, then $\emptyset \not\in S$. However, $\emptyset \subset S$: this is true for any set $S$ whatsoever.

Also, $\emptyset \subseteq \emptyset$, and $S \subseteq S$ for any set $S$.