What Are Sets?

Before we dive into set operations, let's make sure we understand what a set actually is.

Set

A set is a well-defined collection of distinct objects, called elements or members. Sets are usually named with capital letters like $A$ , $B$ , or $C$ .

For example:

$A = 1, 2, 3, 4, 5$ — a set of the first five positive integers
$B = red, blue, green$ — a set of colours
$C = 2, 4, 6, 8, 10$ — a set of even numbers up to 10

Element

An element (or member) is an individual object that belongs to a set. We use the symbol $\in$ to mean "is an element of" and $\in /$ to mean "is not an element of."

Using the set $A = 1, 2, 3, 4, 5$ :

$3 \in A$ means "3 is an element of $A$ " ✓
$7 \in / A$ means "7 is not an element of $A$ " ✓

Analogy

Think of a set like a playlist on your phone. The playlist has a name (the set name), and the individual songs in it are the elements. A song is either on the playlist or it isn't — there are no duplicates, and the order doesn't matter.

Set Notation Essentials

There are several important notations and special sets you need to know:

Universal Set

The universal set, written as $U$ (or sometimes $U$ or $ξ$ ), is the set of all elements being considered in a particular problem. Every other set in the problem is a subset of the universal set.

Empty Set

The empty set (or null set), written as $\emptyset$ or , is the set that contains no elements at all.

Subset

Set $A$ is a subset of set $B$ (written $A \subseteq B$ ) if every element of $A$ is also an element of $B$ .

Other useful notation:

$n (A)$ — the number of elements in set $A$ (also called the cardinality)
Curly braces are used to list the elements of a set

Example

If $U = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10$ and $A = 2, 4, 6$ , then:

$A \subseteq U$ because every element of $A$ is in $U$
$n (A) = 3$ because $A$ has 3 elements
$n (U) = 10$

Warning

Don't confuse $\emptyset$ with $\emptyset$ . The empty set $\emptyset$ has zero elements. The set $\emptyset$ actually has one element — the empty set itself!

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