Question about union with $\{∅\}$: $\{1,2\}\cup \{∅\}=\{1,2,∅\}$? I have two quick questions:
a.
$\{1,2\}\cup \{∅\}=\{1,2,∅\}$ 
is it correct ? 
b.
can I have an explanation why is it not just $\{1,2\}$? Or more importantly why is it important to mention the $∅$ if it is empty? After all $\{1,2\}\cup ∅=\{1,2\}$.
It is a bit unclear to me.Thank you. 
 A: $\varnothing = \{\}$
It is the empty set.   It is not just nothing.   It is a set with no elements.
Sets can be elements of other sets.   The set of the emepty set is not empty, it contains the empty set.   $\{\varnothing\}=\{\{\}\}$
So we have that:
$\{1,2\}\cup\{\} = \{1,2\}$ but $\{1,2\}\cup\{{\small\{\}}\} = \{1,2,{\small\{\}}\}$
Similar to:
$\{1,2\}\cup\{3\} = \{1,2,3\}$ but $\{1,2\}\cup\{{\small\{3\}}\} = \{1,2,{\small\{3\}}\}$
A: $\emptyset$ and $\{\emptyset\}$ are two different things. The first one is the empty set, while the second is a singleton, i.e. a set having a single element, namely the empty set.
Similarly, $\{1,2,\emptyset\}$ has three elements, among which the empty set, while $\{1,2\}$ only has two.
So at the same time, $\emptyset\subset\{1,2,\emptyset\}$ and $\emptyset\in\{1,2,\emptyset\}$.
A: a) It is correct.
b) $\{1,2\}\cup\varnothing=\{1,2\}$. This because $\varnothing$ is empty, i.e. it has no elements. However $\{\varnothing\}$ is not empty. It contains an element, and this element is $\varnothing$. 
Union $\{1,2\}\cup\{\varnothing\}$ is the set containing exactly the elements of $\{1,2\}$, i.e. $1$ and $2$, and the elements of $\{\varnothing\}$, i.e. $\varnothing$. 
Likewise union $\{1,2\}\cup\varnothing$ is the set containing exactly the elements of $\{1,2\}$, i.e. $1$ and $2$, and the elements of $\varnothing$, i.e. no elements at all.
