Empty set

Mathematicians love zero. Ever since its inception around 1770 BC, zero is an important part of mathematical models. The notion of “nothingness”, which zero reflects in the context of counting, is present in all areas. In the context of sets, nothingness is an empty set.

\varnothing = \{\}

Why would we need empty sets? Well, sometimes we want to describe a notion of having no objects under a certain description. For example, since I only watched 3 movies in my life, and all of them were American, I can describe:

\varnothing = \textrm{the set of all non-American movies I've watched}.

A more mathematical example would be something like this:

\varnothing = \{x | x \in \mathbb{N} \textrm{ and } x < 0\}

which says that the set of natural numbers smaller than zero is an empty set. It’s a formal way to say that there are no natural numbers less than zero. Note the use of vertical line |, it is a shorter way to say “where”.