A set is a collection of well-defined objects, called its elements. A set may be given by a condition, as in "the primes at most 10" (set-builder notation), or by listing its elements, as in $\{1, 2, 3\}$ (roster notation). Cantor began the systematic study of sets in the late nineteenth century, and it became the foundation of modern mathematics.

Chapter 1: What Is a Set?

Q: What is the empty set?

The empty set is the set with no elements at all, written $\emptyset$ (or $\{\}$). There is exactly one empty set, and it is a subset of every set. A "set with nothing in it" runs against intuition, but it plays a fundamental role in set operations and in the construction of mathematical objects.

Q: What is the difference between $a \in A$ and $a \notin A$?

$a \in A$ means that $a$ is an element of the set $A$ (that $a$ belongs to $A$), and $a \notin A$ is its negation. For example, with $A = \{1, 2, 3\}$ the statement $2 \in A$ is true, $5 \in A$ is false, and $5 \notin A$ is true. Membership, written $\in$, is the most basic relation symbol in set theory.

The notion that shapes the foundation of mathematics

Introductory (high-school level)

Goal of this page

Understand the intuitive meaning of a set and become comfortable with the notation for elements and membership. Be able to choose between roster notation and set-builder notation.

1. The Notion of a Set

A set is a “collection” of things satisfying some condition. Georg Cantor systematised the notion in the late nineteenth century, and it has become the foundation of modern mathematics.

Definition: a set (intuitive)

A set is a collection of well-defined objects. Each object making up the set is called an element (or member) of that set.

The words “well-defined” carry the weight here. The condition must determine unambiguously whether a given object belongs to the set.

Figure 1. What “well-defined” means. On the left the condition pins the elements down uniquely. On the right the very notion of “large” has no agreed threshold, so it is not settled which numbers belong.

A caveat: not every clear condition yields a set

The account of a set given here is intuitive. It does not follow that any clear condition whatsoever may be used to form a set. The condition “does not contain itself as an element” is not vague at all, yet trying to form a set from it leads to a contradiction. This is taken up in Russell’s paradox, and it is what motivates axiomatic set theory.

Example: some sets

The set of natural numbers from $1$ to $5$
The set of all even numbers
The set of all points on a line $\ell$ in the plane
The set of solutions of the equation $x^2 - 5x + 6 = 0$

2. Membership

When an object $a$ is an element of a set $A$ we write $a \in A$ and say “$a$ belongs to $A$”. When $a$ is not an element of $A$ we write $a \notin A$.

Example: membership

Let $A = \{1, 2, 3, 4, 5\}$.

$3 \in A$ ($3$ belongs to $A$)
$7 \notin A$ ($7$ does not belong to $A$)

Figure 2. The set $A=\{2,4,6,8\}$ drawn as an ellipse. The element $4$ lies inside, so $4 \in A$; the number $5$ lies outside, so $5 \notin A$. The roster notation $\{2,4,6,8\}$ lists exactly what the ellipse encloses.

3. Notation for Sets

3.1 Roster notation (listing)

List every element and enclose the list in braces $\{\}$.

$$A = \{1, 2, 3, 4, 5\}$$

The order does not matter, and repeating an element does not change the set.

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

3.2 Set-builder notation (by condition)

Pick out, from within some set, the elements satisfying a condition. The basic form states explicitly the range over which $x$ varies.

$$B = \{x \in \mathbb{Z} \mid x \text{ is even}\}$$

When the range of $x$ is clear from the context, one often abbreviates this to $\{x \mid P(x)\}$.

Example: set-builder notation

$\{x \in \mathbb{R} \mid x^2 < 4\} = \{x \in \mathbb{R} \mid -2 < x < 2\}$
$\{n \in \mathbb{N} \mid n \text{ is a prime at most } 10\} = \{2, 3, 5, 7\}$

Figure 3. The two notations denote the same set. Listing reads better when there are few elements; when there are many (or infinitely many) a condition is the only option.

4. Special Sets

4.1 The empty set

Definition: the empty set

The set with no elements at all is called the empty set, written $\emptyset$ or $\{\}$.

4.2 Important sets of numbers

Throughout this series $\mathbb{N} = \{1, 2, 3, \ldots\}$, so $0$ is not a natural number. The convention that includes $0$ is equally common, so it is worth checking when reading other texts.

$\mathbb{N}$: the natural numbers $\{1, 2, 3, \ldots\}$
$\mathbb{Z}$: the integers $\{\ldots, -2, -1, 0, 1, 2, \ldots\}$
$\mathbb{Q}$: the rational numbers
$\mathbb{R}$: the real numbers
$\mathbb{C}$: the complex numbers

Figure 4. The number systems are nested: every element of an inner set is also an element of each outer one. The numbers on the right are examples that first appear in that particular ring ($-5$ is an integer but not a natural number, and $\sqrt{2}$ is real but not rational). The symbol $\subset$ is studied in detail in the next chapter.

Summary

Key points of this chapter

Set: a collection of well-defined objects
Membership: $a \in A$ (belongs to), $a \notin A$ (does not belong to)
Roster notation: list the elements, $\{1, 2, 3\}$
Set-builder notation: give a condition, $\{x \mid P(x)\}$
Empty set: the set with no elements, $\emptyset$

Frequently Asked Questions

What is a set?

A set is a collection of well-defined objects, called its elements. It can be written by listing the elements, as in $\{1, 2, 3\}$ (roster notation), or by giving a condition, as in $\{x \mid P(x)\}$ (set-builder notation). Cantor began the systematic study of sets in the late nineteenth century, and it became the foundation of modern mathematics.

What is the empty set?

It is the set with no elements at all, written $\emptyset$ (or $\{\}$). There is exactly one empty set, and it is a subset of every set. A “set with nothing in it” runs against intuition, but it plays a fundamental role in set operations and in the construction of mathematical objects.

What is the difference between $a \in A$ and $a \notin A$?

$a \in A$ means that $a$ is an element of the set $A$, and $a \notin A$ is its negation. For example, with $A = \{1,2,3\}$ the statement $2 \in A$ is true and $5 \notin A$ is true. Membership, written $\in$, is the most basic relation symbol in set theory.