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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.

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

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.

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  • $\begingroup$ Good example hehe, I was thinking in similar here to try understand, but I thought the box couldn't be considered. How wrong I was. Thank you :) $\endgroup$ – Apprentice Jan 4 '15 at 22:34
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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 $'\{1,2\}'$.

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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 $A$.

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  • $\begingroup$ How would they be represented in a Venn diagram? Is it possible? $\endgroup$ – Apprentice Jan 4 '15 at 18:19
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    $\begingroup$ @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. $\endgroup$ – Najib Idrissi Jan 4 '15 at 18:26

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