Set notation

In mathematics, a set is a collection of objects. We can write a set by using braces. Some sets have a special symbol which is used to represent them.

Here are some examples:

• $\{\text{apple}, \text{orange}, \text{pear}\}$ is a set of fruit

• $\{0, 1, 2\}$ is a set containing three particular integers

• $\mathbb{Z}$ is the set of all integers

• $\mathbb{Q}$ is the set of all rational numbers

• $\mathbb{R}$ is the set of all real numbers

• $\{\}$, sometimes written $\emptyset$, is the empty set, the set with no elements

The symbolic notation for “$x$ is in the set of rational numbers” is $x\in\mathbb{Q}$, and similarly for other sets.

Sometimes, we want to say “the set of all numbers which …”; mathematicians use an extension of the brace notation to write this. For example $\{x:x\in\mathbb{R}\ \text{and}\ x^2<4\}$ (read as “the set of $x$ such that $x$ is real and $x^2<4$”) means the set of all real numbers whose square is less than $4$. If it is clear that we are referring to real numbers, this can be abbreviated to $\{x:x^2<4\}$.

A useful related notation is interval notation.