Hi Andrey,

Some of these set theory questions are not obvious.

First of all when you write $\{x\}$ I see that as a set with one element and that element is the letter $x.$ You should start by saying, let $X$ be a set. You then have convinced yourself that $\emptyset \subseteq X.$ You want to know if $\emptyset \in X.$ To prove this is true you would need a mathematical argument to show that the empty set is an element of every set $X.$ To prove this in not true you would need an example of a set $X$ which does not have the empty set as one of its element.

Consider the set $X = \{1, 2 \}.$ This is the set containing two integers, the numbers $1$ and $2.$ Since neither of these is the empty set, $\emptyset \notin X.$

I hope this helps,
Penny