# Why $A=\{1,2\}$ is different of $B=\{\{1,2\}\}$?

If $A=\{1,2\}$ and $B=\{\{1,2\}\}$, why aren't they equals?

I'm really confused with situations where sets are contained in itselves.

• Well, for one thing, the sets do not have the same number of elements (2 vs 1). – mickep Jan 4 '15 at 18:10
• $\{1,2\}$ is a set with two members. $\{\{1,2\}\}$ is a set with only one member. – Michael Hardy Jan 4 '15 at 18:13
• There is no set contained in itself in your example. – user87690 Jan 4 '15 at 18:15
• @user87690 What? I didn't understand that. – Timbuc Jan 4 '15 at 18:20
• @Timbuc: OP said he was confused with situations where sets are contained in themselves. I just noted that there is no such situation here. – user87690 Jan 4 '15 at 18:24

These objects are clearly different and this is the issue here, $\{1,2\}$ is a set containing two elements whereas $\{\{1,2\}\}$ is a set containing a set containing two elements.
Two sets are equal if they have exactly the same elements. But these two have not even one element in common. $\{1,2\}$ has the two elements $'1'$ and $'2\,'$, while $\{\{1,2\}\}$ has only one element $'\{1,2\}'$.
Notice that $B = \lbrace{ A \rbrace}$. So $A \in B$. $1 \notin B$ and $2 \notin B$ but $1,2 \in A$. In other words, the only element of $B$ is $A$.
• @Apprentice Venn diagrams are for different sets included in some bigger set, so for example if your $A, B \subset X$ then you can draw a Venn diagram; you will just have two disjoint disks. – Najib Idrissi Jan 4 '15 at 18:26