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I'm not sure about this expression:\[\emptyset \subset \{\emptyset\}\], I think it's false.

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Its true, i think
This is read: The empty set is a proper subset of the set containing only the empty set.
The empty set is a subset of any set, regardless of whats in it.

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Other answers:

A is a subset of B if and only if for all elements x of A, x is in B. This is vacuously true, since there are no elements x of the empty set. Hence, the empty set is a subset of the empty set
Yeah, I know it's true for the subsets, but I mean the PROPER subset
Proper just means the sets arent equal. So the fact that the empty set is a subset of every set, even though every set isnt empty, means its a proper subset of any non empty set.
Ohh than you @joemath314159 now I got it clear.

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