What exactly is an empty set?

Question

Can you help me with this;

Find the value of $|A|$ if: $A = \{\varnothing\}\cup\varnothing$.

I am having trouble understanding null sets. Is a empty set not a subset of itself? I would be grateful some understanding of this.

Answer 1

The union of $\emptyset$ with any other set is that other set. Therefore, $\{\emptyset\}\cup\emptyset=\{\emptyset\}$. So$$|A|=\bigl|\{\emptyset\}\bigr|=1.$$

Answer 2

We know that set is a well-defined collection of distinct objects (source: https://en.wikipedia.org/wiki/Set_(mathematics))

Here, $\{\emptyset\}$ is a set with an object $\emptyset$ while the empty set $\emptyset = \{\}$. So the union is $\{\emptyset\} \cup \{\} = \{\emptyset\}$ and size is $1$.

In order to prevent confusion, we can simply enumerate (or rename) the object of a set, like we can say "define an object called '$a$', which is the name of the object $\emptyset$ (not the set $\emptyset$)". In this case, our set becomes $\{\emptyset\} = \{a\}$. Rest of this, I'm sure that you can do it much more easily.