Question

We know an empty set is a subset of itself.
However, is an empty set a PROPER subset of itself?

Answer 1

My tutor says for every set, it has at least 2 subsets that are an empty set and the set itself but I doubt that.
He says an empty set has 2 subsets which is an empty set and the set itself which in this case is also and empty set.
Therefore, an empty set is also a proper subset of itself.
Could anybody tell me if it's correct or not?

Answer 2

A set \(A\) is a proper subset of a set \(B\) if \(A\) is a subset of \(B\) but \(A\not=B\), so no, the empty set is not a proper subset of itself.

Answer 3

However (there's always some caveat when it comes to the empty set), the definition of set equality - according to this Wiki page http://en.wikipedia.org/wiki/Subset_and_superset#Definitions states
\[A\not=B\quad``\underbrace{\text{if there exists at least one element of }}B\text{ which is not an element of }A."\]
The bracketed portion is not true for the empty set because it doesn't contain *any* elements. So according to this definition of set equality, the empty set is not a proper subset because the statement that \(\{\}=\{\}\) is vacuously true.

Answer 4

(sorry had to edit something)

Answer 5

hmm so is this statement is incorrect?\[\emptyset \subset \emptyset \]

Answer 6

I would say yes, since \(\emptyset=\emptyset\).

Answer 7

Okay thanks

Answer 8

yw