Set-builder notation ({x | …})

How to read it

There are two ways to write a set in braces. The roster method lists the elements, in curly braces and separated by commas, as in {1, 2, 3}. Set-builder notation gives a rule instead, which suits sets too large to list: OpenStax’s Contemporary Mathematics writes the lowercase alphabet as A = {x | x is a lowercase letter of the English alphabet} and reads it as “the set of all elements x such that x is a lowercase letter of the English alphabet”. Before the bar is the variable, often with the set it is drawn from, as in {x ∈ ℕ : x is even}; after it is the condition (Oscar Levin, Discrete Mathematics).

Also searched as: colon in sets, curly braces in math, roster notation, set builder notation, such that symbol, vertical bar in sets, {x | …}.

A bar or a colon

Levin’s book uses a colon, {x : x > 2}, and notes that some mathematicians use a vertical bar | or the backwards epsilon ∍ instead; OpenStax uses the bar. They mean the same. Unicode lists “such that” among the other names of two characters, the vertical bar ∣ (U+2223, DIVIDES) and ∋ (U+220B, CONTAINS AS MEMBER); between two numbers, the same bar means “divides”, the name Unicode gives that character.

Empty braces

{ } with nothing inside is the empty set, the same as ∅.