Set Notation

What is a Set?

A set is a collection of distinct objects.

Examples from everyday life:

  • A bag of marbles — that’s a set of marbles
  • Your playlist — that’s a set of songs
  • Your contacts list — that’s a set of people

Writing Sets

We use curly braces to write sets:

{1,2,3}\{1, 2, 3\}

This is a set containing 1, 2, and 3.

More examples:

SetDescription
{a,b,c}\{a, b, c\}A set of letters
{2,4,6,8}\{2, 4, 6, 8\}A set of even numbers
{apple,banana}\{\text{apple}, \text{banana}\}A set of fruits

Two Important Rules

Rule 1: No Duplicates

{1,1,2,2,3}={1,2,3}\{1, 1, 2, 2, 3\} = \{1, 2, 3\}

Duplicates are ignored. Each element appears only once.

A set cares about what’s in it, not how many times you wrote it.

Rule 2: Order Doesn’t Matter

{1,2,3}={3,1,2}={2,3,1}\{1, 2, 3\} = \{3, 1, 2\} = \{2, 3, 1\}

A set is just a collection — there’s no “first” or “last” element.

A set cares about what’s in it, not what order you wrote it.

Set Membership

How do we say “this thing is in the set”?

The symbol ∈\in means “is an element of” or “belongs to”:

2∈{1,2,3}2 \in \{1, 2, 3\}

This reads: “2 is in the set {1,2,3}\{1, 2, 3\}”

The symbol ∉\notin means “is NOT an element of”:

5∉{1,2,3}5 \notin \{1, 2, 3\}

This reads: “5 is not in the set {1,2,3}\{1, 2, 3\}”

Practice:

StatementTrue or False?
3∈{1,2,3,4}3 \in \{1, 2, 3, 4\}True
7∈{1,2,3,4}7 \in \{1, 2, 3, 4\}False
a∈{a,b,c}a \in \{a, b, c\}True
d∉{a,b,c}d \notin \{a, b, c\}True

The Empty Set

A set with nothing in it is called the empty set.

Two ways to write it:

  • {}\{\} — curly braces with nothing inside
  • ∅\emptyset — the special empty set symbol

Think of it as an empty bag — still a bag, just nothing inside.

Key facts about the empty set:

  • x∉∅x \notin \emptyset is true for any xx (nothing is in it)
  • There is only one empty set (all empty sets are equal)

Set-Builder Notation

Sometimes listing every element is impossible or tedious.

Instead, we describe a set by a rule:

{x∣x is an even number}\{x \mid x \text{ is an even number}\}

This reads: “The set of all xx such that xx is an even number.”

The vertical bar ∣\mid means ”such that.”

Some books use a colon instead: {x:x>0}\{x : x > 0\}

Examples:

Set-builder notationMeaningAs a list
{x∣x>0}\{x \mid x > 0\}All positive numbers{1,2,3,…}\{1, 2, 3, \ldots\}
{x∣x2=4}\{x \mid x^2 = 4\}Numbers whose square is 4{−2,2}\{-2, 2\}
{n∣n is prime}\{n \mid n \text{ is prime}\}All prime numbers{2,3,5,7,11,…}\{2, 3, 5, 7, 11, \ldots\}

Common Number Sets

Some sets are used so often they have special symbols:

SymbolNameElements
N\mathbb{N}Natural numbers{1,2,3,4,…}\{1, 2, 3, 4, \ldots\}
Z\mathbb{Z}Integers{…,−2,−1,0,1,2,…}\{\ldots, -2, -1, 0, 1, 2, \ldots\}
Q\mathbb{Q}Rational numbersAll fractions ab\frac{a}{b} where b≠0b \neq 0
R\mathbb{R}Real numbersAll numbers on the number line

Why these letters?

  • N\mathbb{N} — Natural
  • Z\mathbb{Z} — Zahlen (German for “numbers”)
  • Q\mathbb{Q} — Quotient (fractions are quotients)
  • R\mathbb{R} — Real

Note: Some definitions include 0 in the natural numbers. We use N={1,2,3,…}\mathbb{N} = \{1, 2, 3, \ldots\} here.