Mathematics often deals not with a single object, but with a collection of objects considered together. Such a collection is called a set, and the objects it contains are its elements.

For example, the colours red, green and blue can be collected into the set

$$A=\lbrace \text{red},\text{green},\text{blue}\rbrace.$$

We write $\text{red}\in A$ because red belongs to the set, while $\text{yellow}\notin A$. The set with no elements is the empty set, written $\varnothing$.

Describing sets

A set can also be described by a property shared by its elements rather than by listing them one by one. For example,

$$B=\lbrace x : x\text{ is a planet in the Solar System}\rbrace$$

describes the set of all planets in the Solar System without listing each one individually.

Subsets

If every element of a set $A$ also belongs to a set $B$, then $A$ is a subset of $B$, written $A\subseteq B$.

Set operations

Sets can be combined. The union $A\cup B$ contains the elements belonging to either set, the intersection $A\cap B$ contains those belonging to both, and the difference $A\setminus B$ contains the elements of $A$ that are not in $B$.

These ideas provide a general language for describing collections and the relationships between them.