# Since the null/empty set is the subset of every set, why is the powerset of {a} different from the powerset of $\{a,\emptyset\}$

The powerset of $\{a\} = \{\{a\},\emptyset\}$ is different from the powerset of $\{a,\emptyset\} = \{\{a\},\emptyset,\{\emptyset\}\}$. Why is that?

I thought the null set was implicitly within ever single set, because the null set is a subset of all of them? Hence, the two sets I gave should be the same and should have the exact same power set.

• The empty set is an element of $\{a,\emptyset\}$. That’s not the case for all sets. – Steve Kass Feb 7 '18 at 2:54
• Ah I see, I've confused subset with element. Thanks – Goldname Feb 7 '18 at 3:05
• Assuming $a\ne\emptyset,$ the set $\{a,\emptyset\}$ has $2$ elements, so its power set will have $4$ elements, not $3.$ The missing element of the power set is $\{a,\emptyset\}$ itself. – bof Feb 7 '18 at 3:17
• The power-set of $\{a.\phi\}$ is $\{\;\phi, \{\phi\},\{a\},\{a,\phi\}\;\}.$.. If $a\ne \phi$ then it has $4$ members. – DanielWainfleet Feb 7 '18 at 4:37

I think you have notation confusion here: $\emptyset = \{\}$, so everytime you see that symbol, you should add an empty set, not an empty place, in your understanding of the set.
So $\{a\}$ implies empty space, not an empty set.
It is the case that $\emptyset \subseteq X$ for every set $X$, but it not the case that $\emptyset \in X$ for every $X$.
In particular, $\{a,\emptyset\} \neq \{a\}$ if $a\neq \emptyset$, as $\emptyset$ is an element of the first, but not the second. Thus, their power sets won't be equal.
• Good point about the word “contains.” To avoid ambiguity, it helps to say things like “$S$ contains $A$ as an element” or “$S$ contains $A$ as a subset.” Or avoid the word “contains” and say “$A$ is a subset of $S$.” or “$A$ is an element of $S$.” – Steve Kass Feb 7 '18 at 17:17