Why $\{\emptyset\} \not \subset\{\{\emptyset\}\}$?

Asked Feb 06 '14 at 20:02

Active Feb 06 '14 at 20:24

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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/} }

edited Feb 06 '14 at 20:24

jimjim

9,855

asked Feb 06 '14 at 20:02

GoofyBall

1

I think that this was covered at least four times before on the site. – Asaf Karagila Feb 06 '14 at 20:04
Close enough duplicates: http://math.stackexchange.com/questions/135218/set-theory-subset-of-set/ and http://math.stackexchange.com/questions/638560/why-can-a-1-element-set-be-a-member-of-another-set-but-not-a-subset-of-it (and there are exact duplicates too) – Asaf Karagila Feb 06 '14 at 20:05
@AsafKaragila, a link would greatly be appreciated then! – GoofyBall Feb 06 '14 at 20:05
What exactly do you mean by { }? – GPerez Feb 06 '14 at 20:07
{ } can be taken to mean a set, all of whose elements have been removed. – GoofyBall Feb 06 '14 at 20:08
So what's the difference between that and $\emptyset$? – GPerez Feb 06 '14 at 20:08
1

Note, by the way, that ${\varnothing}$ is not empty. It has an element. – Asaf Karagila Feb 06 '14 at 20:08
@GPerez, the author of the textbook I am reading gives the subset {} as a subset of any set. – GoofyBall Feb 06 '14 at 20:15
I fixed up the latex for title, can you fix up the rest please? – jimjim Feb 06 '14 at 20:25

Why $\{\emptyset\} \not \subset\{\{\emptyset\}\}$?

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