Is $∅ ∈ \{\{∅\}\}$ true or false?
I think it is false and my reasoning is this,
∅ is an element of a set of subset ∅
Since ∅ is an element of the set, it is therefore not an element of the subset inside the set. Am I right?
Is ∅ ⊄ {∅,1,2} true or false?
I think it is true and my reasoning is this,
∅ is not a subset of set ∅, 1, 2
∅ is an empty set therefore it is a subset of {∅, 1, 2}
Lastly, is it right to say that in any power set of X, ∅ will always either be ∈ or ⊆ of power set X?
Best Answer
The emptyset, which we'll denote $\varnothing$, is a subset of every set, and it is even a subset of the empty set itself.
$\varnothing \subset \{\{\varnothing\}\}$ but $\varnothing \notin \{\{\varnothing\}\}$. Our set has one single element, $\{\varnothing\}$, the set containing the emptyset, so $\varnothing$ is not an element of the initial set.
However $\{\varnothing\} \in \{\{\varnothing\}\}$.
And $\{\{\varnothing\}\} \subseteq \{\{\varnothing\}\}$.
In your second example, let's call $S= \{\varnothing, 1, 2\}$.
Then $\varnothing \subseteq S$, because the emptyset is a subset of every set. $\varnothing \in S$, as well, as it is an element of S. And $\{\varnothing\} \subseteq S$, just like $\{1, 2\} \subseteq S$.
So the powerset of S, the set containing all subsets of S is as follows:
$$\{\varnothing, \{\varnothing\}, \{\varnothing, 1,\}, \{\varnothing, 2\}, \{1\}, \{2\}, \{1, 2\}, \{\varnothing, 1, 2\}\}$$