Element Inclusion vs. Subsets

When an element $n$ is a part of a set $S$, we write: $n \in S$ (

$n \in S$

), which is read as "$n$ is in $S$", "$n$ belongs to $S$", or "$n$ is a part of $S$".

When an element $n$ is not in the set $S$, we write: $n \not\in S$ (

$n \not\in S$

).

Examples: Let $S = \{$😀$,$ 😁$,$ 😎$,$ 😜$\}$. We see that 😁 $\in S$, while 🤗 $\not\in S$.

---

When you have two sets, $A$ and $B$, and all the elements of $A$ are included in $B$, we say that $A$ is a subset of $B$. We write: $A \subseteq B$ (

$A \subseteq B$

). Also, if $B$ has more unique element than $A$, we say that $A$ is a proper subset of $B$, and write: $A \subset B$ (

$A \subset B$

).

If at least one of the items of $A$ is not in $B$, then $A$ is not a subset of $B$, so we write $A \not\subseteq B$ or $A \not\subset B$ (

$A \not\subseteq B$

or

$A \not\subset B$

).