What a set is, and how to write one

A set is a well-defined collection: for any object, it is clear whether it belongs. A set is written in roster form by listing its elements, or in set-builder form by stating the rule they satisfy.

A well-defined collection

A set is a collection of objects that is well-defined. Well-defined means: for any object whatever, it can be decided without doubt whether the object is in the collection or not.

“The vowels of the English alphabet” is a set. It contains a, e, i, o and u, and every other letter is definitely not in it. “The tall students in the class” is not a set, because there is no agreed height at which a student becomes tall. Two people could decide differently about the same student.

Elements and the membership symbol

The objects in a set are its elements, or members. Sets are usually named with capital letters, such as A, B or V. Elements are usually written with small letters.

If a is an element of A, we write a ∈ A, read “a belongs to A”. If a is not an element of A, we write a ∉ A. For V, the set of vowels, e ∈ V and b ∉ V.

Roster form

In roster form, also called tabular form, the elements are listed inside curly brackets, separated by commas:

V = {a, e, i, o, u}

Two conventions apply. Order does not matter: {a, e, i, o, u} and {u, o, i, e, a} are the same set. Each element is written once: the set of letters in “school” is {s, c, h, o, l}. A set records which objects belong, not how many times they appear.

Set-builder form

A set with many elements is long to list, and an infinite set cannot be listed completely. Set-builder form states the property that the elements, and only the elements, have:

V = { x : x is a vowel of the English alphabet }

The colon is read “such that”. This reads: “the set of all x such that x is a vowel of the English alphabet”.

{ x : x is an even natural number, x < 10 } is the same set as {2, 4, 6, 8}. { x : x is an even natural number } is infinite; its roster form can only be suggested, as {2, 4, 6, …}.