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.

• What two conditions are necessary for $A\subset B$? – peoplepower Dec 27 '12 at 0:52
• Ohh I see that is false because $\emptyset = \emptyset$ – Victor Jose Arana Rodriguez Dec 27 '12 at 0:55
• Now I see that it was a dumb question sorry. – Victor Jose Arana Rodriguez Dec 27 '12 at 0:56
• Pay attention to the fact that many authors use the symbol $\,\subset\,$ to denote weak containment. – DonAntonio Dec 27 '12 at 0:58
• @VictorJoseAranaRodriguez: You can post that as an answer to the question and accept it, if you like. – Ben Millwood Dec 27 '12 at 1:20

$A$ 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 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: {∅}

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.

• $B \setminus A \neq \varnothing$ does not imply that $A$ is a subset of $B$. (But, if it is, then it is indeed a proper subset.) – François G. Dorais Dec 27 '12 at 4:41
• Thanks for the comment. I actually meant an implication instead of a bi-implication. – Haskell Curry Dec 27 '12 at 4:52

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.

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.

i think empty set is not a proper subset of itself because ∅=∅ .

i think empty set is not a proper subset of any non empty set because empty set has no any elment.