Math 330 - Comments

Membership vs. Subset

Let $A$ and $B$ be sets whose elements come from some set $X$.

$x\in A$ means $x$ is an element (or member) of $A$.

$x\notin A$ means $x$ is not an element of $A$.

$A\subseteq B$ means $A$ is a subset of $B$; that is, if $x\in A$, then $x\in B$.

Formally written: $\forall x\in X,$ if $x\in A$, then $x\in B$.

$A\not\subseteq B$ means $A$ is not a subset of $B$.

Formally written: $\exists x\in X$ such that, $x\in A$ and $x\notin B$. (N.B. You should agree with this definition, see Section 3.3 on Negations.)

Examples using the symbols '$\in$' and '$\notin$':

$1\in \{1\}$.

$3\notin \{1\}$.

$1\in \{1,2\}$.

$1\in \mathbb{N}$.

$0\notin \mathbb{N}$.

$0\in \mathbb{Z}_{\geq 0}$.

$\spadesuit\in \{\heartsuit, \spadesuit, \diamondsuit, \clubsuit\}$.

$\omega\notin \{\heartsuit, \spadesuit, \diamondsuit, \clubsuit\}$.

$\{0\}\in \{\{0\},\heartsuit\}$.

$0\notin \{\{0\},\heartsuit\}$.

$1\notin \varnothing$.

$\varnothing\notin \varnothing$.

$\varnothing\in \{\varnothing\}$.

$\{\varnothing\}\notin \{\varnothing\}$.

$\varnothing\in \{\varnothing,\{\varnothing\}\}$.

$\{\varnothing\}\in \{\varnothing,\{\varnothing\}\}$.

Examples using the symbol '$\subseteq$' and '$\not\subseteq$':

$\{1\}\subseteq \{1\}$.

$\{3\}\not\subseteq \{1\}$. (This is because there exists an element of $\{3\}$, namely $3$, that is not an element of $\{1\}$.)

$\{1\}\subseteq \{1,2\}$.

$\{1\}\subseteq \mathbb{N}$.

$\{0\}\not\subseteq \mathbb{N}$.

$\{0\}\subseteq \mathbb{Z}_{\geq 0}$.

$\{\spadesuit\}\subseteq \{\heartsuit, \spadesuit, \diamondsuit, \clubsuit\}$.

$\{\omega\}\not\subseteq \{\heartsuit, \spadesuit, \diamondsuit, \clubsuit\}$.

$\{\{0\}\}\subseteq \{\{0\},\heartsuit\}$.

$\{0\}\not\subseteq \{\{0\},\heartsuit\}$.

$\{1\}\not\subseteq \varnothing$.

$\varnothing\subseteq \varnothing$.

Proposition 5.7. For any set $A$, $\varnothing\subseteq A$.

$\varnothing\subseteq \{\varnothing\}$.

$\{\varnothing\}\subseteq \{\varnothing\}$.

$\varnothing\subseteq \{\varnothing,\{\varnothing\}\}$.

$\{\varnothing\}\subseteq \{\varnothing,\{\varnothing\}\}$.

$\{\{\varnothing\}\}\subseteq \{\varnothing,\{\varnothing\}\}$.

$\{\{\{\varnothing\}\}\}\not\subseteq \{\varnothing,\{\varnothing\}\}$.