First sorry about my English, I seek on all internet and Sheinerman's book about this examples:

1.{{∅}} ⊂ {∅,{∅}}?

2.{{∅}} ∈ {∅,{∅}}?

3.{{∅}} ⊂ {∅,{{∅}}}?

4.{{∅}} ∈ {∅,{{∅}}}?

5.{{∅}} ⊂ {∅,{∅,{∅}}}?

6.{{∅}} ∈ {∅,{∅,{∅}}}?

7.{∅} ⊂ {∅,{∅}}?

8.{∅} ∈ {∅,{∅}}?

I found a similar answer in the below link, but I can't understand [question] : Is an empty set equal to another empty set? For me, all obvious answers for "⊂" must be yes, because the empty set is a subset of every set,but in the link example :

(n) {{∅}}⊂{{∅},{∅}} [False], this is shaking my head.

Can someone help ? Thanks

  1. Is $\{\{\emptyset\}\}\subset \{\emptyset,\{\emptyset\}\}$? You need to check for each element of $\{\{\emptyset\}\}$ (there is only one such element, namely $\{\emptyset\}$) whether it is en element of $\{\emptyset,\{\emptyset\}\}$. This set has precisely two elements, $\emptyset$ and $\{\emptyset\}$, and the latter is in deed the one we look for. So: YES

  2. NO, for $\{\{\emptyset\}\}$ is not among the two elements $\emptyset$ and $\{\emptyset\}$ of the right hand side.

  3. NO, for the only element $\{\emptyset\}$ of the set on the left is not among the two elements $\emptyset$ and $\{\{\emptyset\}\}$ on the right

  4. YES, for $\{\{\emptyset\}\}$ itself is among the two elements $\emptyset$ and $\{\{\emptyset\}\}$ on the right

You should be able do solve 5-8 similarly (and should arrive at two NO and two YES).

However, since $\{\{\emptyset\},\{\emptyset\}\}$ has only one element $\{\emptyset\}$ (even though it looks like having two elements), statement $(n)$ is true if $\subset$ allows improper inclusion (which is often written as $\subseteq$) and false if $\subset$ denotes proper inckusion (often written less ambiguously as $\subsetneq$)

  • $\begingroup$ en $\mapsto$ an $\;$ $\endgroup$ – user57159 Mar 15 '15 at 7:16
  • $\begingroup$ Also "incklusion" should be "ink illusion" or "inclusion" on the last line of the text. Probably the latter. $\endgroup$ – Asaf Karagila Mar 15 '15 at 10:13

The symbol $\subset$ is used in some places for "proper inclusion", which means $A\subset B$ if $A$ is a subset of $B$ and $A\neq B$ (or $A$ is a proper subset of $B$); but in other places it is used for "improper inclusion" which allows the option for $A=B$, so $A$ is not a proper subset of $B$, just a subset.

Since $\{\{\varnothing\}\}=\{\{\varnothing\},\{\varnothing\}\}$, the inclusion is not proper. But there is still inclusion, if you consider improper inclusion, better written as $A\subseteq B$.

I will also add that it is not true that all the truth values of the statements with $\subset$ in your questions are "obviously true", since $\{\{\varnothing\}\}$ is not the empty set, it is a set which has a single element, and that element is a set which has a single element again, and that is the empty set. It is a box containing a box containing a box containing nothing.

Instead, it is better to check the truth values more carefully.


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