A set groups well-defined objects; membership, inclusion and operations describe its structure.

Sets

Definition

A set is a collection of well-defined objects called its elements. For example, we write

$A=\{2,4,6\}$

and $4\in A$ to say that 4 belongs to $A$ . Conversely, $5\notin A$ .

Order and repetitions do not change a set: $\{2,4,6\}$ , $\{6,4,2\}$ and $\{2,2,4,6\}$ denote the same set. Only one question matters: which elements belong to the collection?

Describing a set

Its elements can be listed by extension:

$B=\{a,e,i,o,u\}$

or defined by a property:

$C=\{n\in\mathbb{N}\mid n<5\}=\{0,1,2,3,4\}$

Here we use the convention $0\in\mathbb{N}$ . The vertical bar is read “such that.” The property must decide unambiguously whether an object belongs to the set.

Empty set, equality and cardinality

The empty set, written $\varnothing$ , contains no element. It must not be confused with $\{\varnothing\}$ , which contains one element: the empty set itself.

Two sets are equal if they have exactly the same elements. For a finite set $A$ , its cardinality $|A|$ is its number of elements. Thus, if $A=\{2,4,6\}$ , then $|A|=3$ .

Inclusion and subsets

We write $A\subseteq B$ when every element of $A$ also belongs to $B$ . For example,

$\{2,4\}\subseteq\{1,2,3,4\}$

Membership and inclusion are not the same relation: $2\in A$ compares an object with a set; $A\subseteq B$ compares two sets.

Every set is included in itself, and the empty set is included in every set.

Three fundamental operations

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

Operation	Meaning	Result
Union $A\cup B$	Elements in $A$ or in $B$	$\{1,2,3,4\}$
Intersection $A\cap B$	Elements shared by $A$ and $B$	$\{3\}$
Difference $A\setminus B$	Elements of $A$ that are not in $B$	$\{1,2\}$

The word “or” in a union is inclusive: an element present in both sets does belong to the union, but appears there only once.

If a universal set $U$ is fixed, the complement of $A$ in $U$ is $U\setminus A$ . Without an explicitly chosen universe, “the complement of $A$ ” is incomplete.

Diagrams help, but do not define

A Venn diagram represents sets as regions. Overlap visualizes intersection, and the combined regions visualize union. It is a reasoning aid, not the definition: a proof must return to membership conditions.

To show $A\cap B\subseteq A$ , take an arbitrary element $x\in A\cap B$ . By definition of intersection, $x\in A$ and $x\in B$ ; therefore $x\in A$ . The inclusion is proved.

Why the abstraction is useful

Sets provide a common vocabulary for mathematics and computer science. A domain of values, query results, graph vertices or cases accepted by a rule can all be treated as sets. Relations then become sets of ordered pairs, and functions rely on a domain and a codomain.

This idea is therefore a direct prerequisite for studying relations, functions, databases, probability and graphs.

Summary

A set is determined by its elements: membership connects an object to a set; inclusion compares two sets; union, intersection and difference build new sets.

Educational references