What are the sub-sets of a null set? I don't get any other set than {}
. Please help me out. Thanks.
What are the sub-sets of a null set?
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$\begingroup$
elementary-set-theory
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0@Qiaochu Fair enough. I hadn't heard of that phrase used in that way before. – 2011-06-13
2 Answers
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You are right: the empty set has precisely one subset: the empty set.
As a formula: $P(\emptyset)=\{\emptyset\}=\bigl\{\{\}\bigr\}$.
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3To 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. – 2011-06-13
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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 $\emptyset$.