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.