Sets

Your favourite songs, the even numbers, every country with a coastline: as soon as you gather things into a collection, you are working with a set. Mathematics gives this simple idea a clear language that can order surprisingly much.

What is a set?

A set is a collection of distinguishable objects, its elements. You write it in curly braces, for example $A = \{1, 2, 3\}$. That 2 belongs to the set is written $2 \in A$, read as 2 is an element of A. If an object does not belong, you write $5 \notin A$. Order does not matter, and each element counts only once.

Subsets

If every element of one set also sits in another, the first is a subset of the second. You write $\{1, 2\} \subseteq \{1, 2, 3\}$. The symbol $\subseteq$ is read as is a subset of. The smaller set is then fully contained in the larger one. Since this still holds when both sets are equal, every set is a subset of itself.

Union and intersection

From two sets you can build new ones. The union $A \cup B$ contains all elements lying in $A$ or in $B$, and likewise those sitting in both. Such a shared element still appears only once in the union. The intersection $A \cap B$ contains only the elements sitting in both at once. The small $\cup$ looks like a container that gathers everything, the $\cap$ like a roof under which only the shared part fits.

The empty set

If two sets share no element, their intersection is empty. This special set with no element at all is called the empty set and is written $\emptyset$ or $\{\,\}$. It is a subset of truly every set and shows up whenever nothing remains.