Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

What are the sub-sets of a null set? I don't get any other set than {}. Please help me out. Thanks.

share|improve this question
Sanity check: a null set has $0$ elements, so its powerset had better have $2^0 = 1$ elements. –  Qiaochu Yuan Jun 13 '11 at 14:23
@fahad: For such a little guy, the empty set sure can cause a lot of trouble! Visualize a set as a plastic bag, with "things" in it. One particularly simple set is the empty plastic bag. –  André Nicolas Jun 13 '11 at 14:43
@fahad: And earlier, I forgot to mention. A plastic bag can have other plastic bags in it, so some or all of the elements of a set can be themselves sets. –  André Nicolas Jun 13 '11 at 15:50
Perhaps he means "null set" in the sense of measure theory, that is: a null set is a set of measure zero. Not at all the same thing as the empty set. I suspect this because he says a null set. –  GEdgar Jun 13 '11 at 17:05
@Qiaochu I don't see how the author's slip here implies anything about his sanity. –  Doug Spoonwood Jun 13 '11 at 20:56
show 2 more comments

2 Answers

up vote 14 down vote accepted

You are right: the empty set has precisely one subset: the empty set.

As a formula: $P(\emptyset)=\{\emptyset\}=\bigl\{\{\}\bigr\}$.

share|improve this answer
Thanks a lot. Short and understandable answer. –  Fahad Uddin Jun 13 '11 at 15:01
To emphasize a subtlety in Rasmus's answer, the power set of the empty set is NOT the empty set (i.e. $\emptyset$ or $\{\}$). It is the set CONTAINING the empty set (i.e. $\{\emptyset\}$ or $\{\{\}\}$). The difference is subtle and easy to overlook. –  Austin Mohr Jun 13 '11 at 15:46
add comment

I would like to use the definition of "subsets".

Definition: Let $A$,$B$ be sets. We say that $A$ is a subset of $B$, denoted $A\subset B$, iff every element of $A$ is also an element of $B$, i.e.
For any object $x$, $x\in A\Rightarrow x\in B$.

Now assume that $A\subset\emptyset$, i.e., for any object $x$, $x\in A\Rightarrow x\in \emptyset$. Since $x\in\emptyset$ is always false, $x\in A$ should also be always false. And thus $A$ has to be the empty set $\emptyset$.

share|improve this answer
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.