# Why is $\{\{1\}\}$ not equal to $\{1,\{1\}\}$?

Determine whether each of these pairs of sets are equal$$A = \{\{1\}\} \qquad \qquad B = \{1, \{1\}\}$$

I believe $A$ is equal to $B$ because all elements in $A$ are in $B$, but the answer says that it's not.

• A has one element but B has two. Jul 16, 2016 at 2:02
• Are you asking if A is a subset of B or if A is equal to B? Jul 16, 2016 at 2:06
• If being a subset is the same as being equal, then all sets are empty. Jul 16, 2016 at 2:11
• Aren't A a subset of B the same as A equal B? What ?!?!?! absolutely of course not! A equal B means A and B have the same elements. A subset B means all of the elements of A are in B but not nescessarily are all elements of B in A. In the is case they can't be equal because they don't have the same elements. B has 1 as an element but A does not. $B \not \subset A$. For $A=B$ then both $A\subset B$ and $B \subset A$. Jul 16, 2016 at 2:35
• The answer actually depends on whether $1=\{1\}$ or $1\ne\{1\}$. The former would make $1$ a Quine atom, which is an entity you will not want to encounter in the set theory you learn. Jul 16, 2016 at 8:04

Think of $A$ as a bag which contains within it another smaller bag with a one in it.

$A=\underbrace{\{~~~~~~~\overbrace{\{1\}}^{\text{second bag}}~~~~~~~~\}}_{\text{first bag}}$

On the other hand, $B$ is a bag which contains in it not only a second bag with a one in it, but also a one which is loose.

$B=\underbrace{\{~~~~~~~~\overbrace{\{1\}}^{\text{second bag}}~~~~~\overbrace{1}^{\text{this too}}~~~~~~~\}}_{\text{first bag}}$

$1\in B$ but $1\not\in A$. There is no "loose 1" in $A$, there is only a bag with a one in it in $A$.

Thus, $A\neq B$

You're correct that all elements in $A$ are in $B$, but not the other way around - $B$ includes the element $1$, but $A$ only has $\{1\}$. Think of it like boxes - $B$ is a box that includes one item and also a box that itself contains one item; $A$ is just a box containing a box containing an item.

• And now think about this: does the box of all the boxes that don't contain themselves contain itself? Hahaha Jul 16, 2016 at 11:15