Empty Sets question

Question

I need some help understanding empty sets being elements vs subsets to another set.

This ask if it is true or false.

{{∅}} ∈ {∅,{∅}}

{{∅}} ⊆ {∅,{∅}}

I know that they are both false but could someone explain how so?

Answer 1

$\{\emptyset,\{\emptyset\}\}$ has precisely two elements, $\emptyset$ and $\{\emptyset\}$, but $\{\{\emptyset\}\}$ is not one of them. So $\{\{\emptyset\}\}\notin \{\emptyset,\{\emptyset\}\}$.

On the other hand, each element of $\{\{\emptyset\}\}$ (that is, its only element $\{\emptyset\}$) is an element of $\{\emptyset,\{\emptyset\}\}$. So $\{\{\emptyset\}\}\subseteq \{\emptyset,\{\emptyset\}\}$.

Answer 2

just replace the symbol for empty set with your favourite letter and regard the brackets as part of the name of the element.

For the first example the two elements are $x$ and $\{x\}.$ But the RHS is neither of these since it is $\{\{x\}\}.$ So it is not in the set.

Answer 3

Does this help? $$\begin{array}{|c:c|}{\begin{array}{rl}\color{darkblue}\emptyset \in& \big\{\color{darkblue}\emptyset, \{\color{green}\emptyset\}\big\} \\[0.5ex] \{\color{green}\emptyset\}\in&\big\{\color{darkblue}\emptyset,\{\color{green}\emptyset\}\big\} \\[1.5ex] \color{darkred}\emptyset\subseteq& \big\{\color{darkblue}\emptyset,\{\color{green}\emptyset\}\big\} \\[0.5ex] \big\{\color{darkblue}\emptyset\big\}\subseteq& \big\{\color{darkblue}\emptyset,\{\color{green}\emptyset\}\big\} \\[0.5ex] \big\{\{\color{green}\emptyset\}\big\}\subseteq& \big\{\color{darkblue}\emptyset,\{\color{green}\emptyset\}\big\} \\[0.5ex] \big\{\color{darkblue}\emptyset, \{\color{green}\emptyset\}\big\}\subseteq& \big\{\color{darkblue}\emptyset,\{\color{green}\emptyset\}\big\}\end{array}}&{ \begin{array}{rl}\color{darkblue}A \in& \big\{\color{darkblue}A, \{\color{green}B\}\big\} \\[0.5ex] \{\color{green}B\}\in&\big\{\color{darkblue}A,\{\color{green}B\}\big\} \\[1.5ex] \color{darkred}{\{\}}\subseteq&\big\{\color{darkblue}A,\{\color{green}B\}\big\} \\[0.5ex] \big\{\color{darkblue}A\big\}\subseteq&\big\{\color{darkblue}A,\{\color{green}B\}\big\} \\[0.5ex] \big\{\{\color{green}B\}\big\}\subseteq&\big\{\color{darkblue}A,\{\color{green}B\}\big\} \\[0.5ex] \big\{\color{darkblue}A, \{\color{green}B\}\big\}\subseteq&\big\{\color{darkblue}A,\{\color{green}B\}\big\}\end{array}}\end{array}$$

Answer 4

The two statements are NOT both false. The first one, {{∅}} ∈ {∅,{∅}}, is indeed false, but the second one, {{∅}} ⊆ {∅,{∅}}, is true.

A ∈ B means A is an element of B and A ⊆ B means A is a subset of B.

∅ is no different from {}, so equivalently you want to see if {{{}}} is an element of {{},{{}}} and if {{{}}} is a subset of {{},{{}}}.

First, you need to figure out all the elements of the set {{},{{}}}. You can just remove the leftmost and the rightmost curly braces and you get {}, {{}} - meaning that the elements are {} and {{}}. Since {{{}}} is neither of them, {{∅}} ∈ {∅,{∅}} is false.

For the second statement, you should figure out all the subsets of the set {{},{{}}}. You may want to iterate all possible combinations of its elements - {} and {{}}.

1. The subset can have neither of {} and {{}}, so the subset can be an empty set, namely ∅ or {};

2. The subset can have only {}. At this point, you may want to add curly braces around {}, because it is a set containing {}. Hence, the subset can be {{}};

3. The subset can have only {{}}, the other element of {{},{{}}}. Similarly to 2, the subset can be {{{}}}, which proves {{∅}} ⊆ {∅,{∅}} to be true;

4. We may continue to get the last subset, the set containing both elements or simply {{},{{}}} itself.