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I'm not sure about if this expression is true or false. $\emptyset \subset \emptyset $. I mean proper subset.

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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

7 Answers 7

up vote 3 down vote accepted

$A$ 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.

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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.

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$B \setminus A \neq \varnothing$ 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

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: {∅}

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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.

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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.

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

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

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