# Why $\{\emptyset\} \not \subset\{\{\emptyset\}\}$? [duplicate]

In my text book it is written that:

{ } ⊆ { }; { } ⊆ {0/}; { } ⊆ { {0/} }; { } ⊆ C; and {0/} ⊆ C; { {0/} } ⊆ C; but {0/} is not a subset of { {0/} } since the only element of {0/} is 0/ and the only element of { {0/} } is {0/}, so the element of {0/} is not an element of { {0/} }.

A set with no elements is an empty set, denoted by {0/}.

There are three parts to my question.

Firstly, what is the distinction between {0/} and { }? Why is the latter not an empty set like the former?

Secondly, is { {0/} } an element and at the same time a set?

And finally, howcome is {0/} is not a subset of { {0/} }? How can an empty set not be a subset of an empty set?

This is my first question on math.stackexchange: I have no formal Mathematics background so please don't presume too much as much as I want to learn...

Thank you, internet!

UPDATE: Let C = { 0/, {0/} }

• I think that this was covered at least four times before on the site. – Asaf Karagila Feb 6 '14 at 20:04
• Close enough duplicates: math.stackexchange.com/questions/135218/… and math.stackexchange.com/questions/638560/… (and there are exact duplicates too) – Asaf Karagila Feb 6 '14 at 20:05
• @AsafKaragila, a link would greatly be appreciated then! – GoofyBall Feb 6 '14 at 20:05
• What exactly do you mean by { }? – GPerez Feb 6 '14 at 20:07
• { } can be taken to mean a set, all of whose elements have been removed. – GoofyBall Feb 6 '14 at 20:08