Why can a one-element set (a singleton) be a member of another set but not a subset of it?

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I have came across this in a textbook:

$\{2\}\nsubseteq\{\{2\}\}$ but $\{2\}\in\{\{2\}\}$

I understand that $\{2\}$ is an element (member) of the other set but considering $\{2\}$ is a set itself, why is it not a subset?

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2 Answers

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The assertion $\{ 2 \} \subseteq \{ \{ 2 \} \}$ states that every element of $\{2\}$ is an element of $\{\{2\}\}$. But $2$ is the only element of $\{2\}$, and $2$ is not an element of $\{\{2\}\}$ because the only element of this latter set is $\{2\}$, and $2 \ne \{2\}$!

Just because something is a set and it's an element of another set doesn't mean it's a subset of it.

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To phrase it differently:

1. $\{\{2\}\}$ is a set containing a set
2. $\{2\}$ is a set containing a number

Since they are sets containing different kind of things, they cannot have a superset-subset relation to each other...

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