# $\{\{\{∅\}\}\} ⊂ \{∅, \{∅\}\}$ true or false?

I know $\{\{∅\}\} ⊂ \{∅, \{∅\}\}$ is true because the right hand side contains the left hand side and another element, at least according to the text I'm going by so wouldn't that make $\{\{\{∅\}\}\} ⊂ \{∅, \{∅\}\}$ true?

• Is $\{\{ \emptyset \}\}$ an element of the right hand side? May 27 '17 at 22:10
• If we colour the outermost parentheses, do you think it's a bit more clear? $\color{red}{\{} \{\{ \varnothing \}\} \color{red}{\}}$ $\color{red}{\{} \varnothing, \{\varnothing\}\color{red}{\}}$ May 27 '17 at 22:12
• I'm not sure. Maybe if in {∅, {∅}} you ignore ∅ you can look at it as {{∅}}. I'm not certain if that's possible
– Ryan
May 27 '17 at 22:13
• We still wouldn't have $\{\{\{∅\}\}\} ⊂ \{\{∅\}\}$. May 27 '17 at 22:17
• I do agree that all these curly brackets and empty sets can be confusing to navigate, and apart from "be careful and if all else fails, use the definitions directly" there isn't really any tip I can give you. May 27 '17 at 22:20

A set $A$ is a subset of a set $B$ if every element of $A$ is an element of $B$. If $A = \{\{\{\emptyset\}\}\}$ and $B = \{\emptyset, \{\emptyset\}\}$ then the elements of $A$ are $\{\{\emptyset\}\}$ and the elements of $B$ are $\emptyset$ and $\{\emptyset\}$. So $A \not\subseteq B$ because $\{\{\emptyset\}\} \in A$ but $\{\{\emptyset\}\} \notin B$. Note that there is a difference between the element $\{ \emptyset \}$ of $B$ and the element $\{\{\emptyset\}\}$ of $A$.