Is empty set a proper subset of itself? I'm not sure about if this expression is true or false. $\emptyset \subset \emptyset $. I mean proper subset.
 A: $A$ is a proper subset of $B$ if $A\subseteq B$ but not $A\supseteq B$. For any set $X$ we have $X\subseteq X$ and therefore also $X\supseteq X$.
Thus no set is a proper subset of itself, and neither is the empty set.
A: If $ A $ is a proper subset of $ B $, then $ B \setminus A \neq \varnothing $. Hence, if $ \varnothing $ were a proper subset of $ \varnothing $, then $ \varnothing \setminus \varnothing \neq \varnothing $, which is not true.
A: A set is a collection of objects of some kind.  We can identify the objects in a set by listing them (if the set is finite) or by giving a rule that tells us whether an object is in the set (all positive integers less than 10, or all even integers).  A set can contain one or more sets, for example the set of all the different sets that can contain zero or more of the integers 1 and 2 (try it, there are only four possible sets)
The empty set is a set that contains no objects, not even the empty set (considered as an object that could be in the set.  So the empty set cannot be contained in itself.  A set containing the empty set could be written by explicitly listing its contents: {∅}
A: If $A$ is a proper subset of $B$ then $A\neq B$. Every mathematical object is equal to itself, and so is the empty set. 
Do note that $\subset$ is not always used for proper inclusion, which is why you might see $\varnothing\subset\varnothing$ written in some places. 
A: No. Let $x\in\emptyset$; since this is false, it implies $x\in\emptyset$.  Hence
we have $\emptyset = \emptyset$.  This rules out the empty set being a proper subset of itself.
A: i think empty set is not a proper subset of any non empty set because empty set has no any elment.
A: i think  empty set is not a proper subset of itself because ∅=∅ .
