What is the union of two sets?

The union A ∪ B contains all elements belonging to A or B or both. It uses inclusive 'or' — elements in both sets appear once. Union is commutative and associative.

What is the intersection of two sets?

The intersection A ∩ B contains only elements belonging to both A and B. If A ∩ B = ∅, the sets are disjoint. Intersection is commutative and associative.

What is the complement of a set?

The complement Aᶜ contains all elements in the universal set U that are not in A. Taking the complement twice returns the original set: (Aᶜ)ᶜ = A.

What is set difference?

The set difference A \ B contains elements in A but not in B. Unlike union and intersection, set difference is not commutative — A \ B ≠ B \ A in general.

What is symmetric difference?

The symmetric difference A △ B contains elements in exactly one of the two sets — either A or B, but not both. It equals (A \ B) ∪ (B \ A).

What does disjoint mean for sets?

Two sets are disjoint when they have no elements in common — their intersection is the empty set: A ∩ B = ∅.

How does set difference relate to complement?

Set difference can be expressed using complement: A \ B = A ∩ Bᶜ. The complement Bᶜ equals U \ B where U is the universal set.

Is union commutative?

Yes. A ∪ B = B ∪ A for any sets A and B. Union is also associative: (A ∪ B) ∪ C = A ∪ (B ∪ C).

A ∩ Aᶜ = ∅. No element can belong to both A and its complement simultaneously. Similarly, A ∪ Aᶜ = U (the universal set).

What is the symmetric difference of a set with itself?

A △ A = ∅. Since symmetric difference includes elements in exactly one set, there are no elements in A but not A. Also, A △ ∅ = A.

Operations on Sets

Summary: All Five Operations at a Glance

Combining and Manipulating Sets

Set operations combine sets to form new sets. These operations — union, intersection, complement, difference, and symmetric difference — provide the tools for manipulating collections and form the basis for the algebraic rules that govern set theory. Visualizing these operations through Venn diagrams makes their behavior immediately clear.

Key Terms

Union— elements in at least one of the sets

Intersection— elements common to both sets

Complement— elements in the universal set but not in the given set

Set Difference— elements in one set but not the other

Symmetric Difference— elements in exactly one of two sets

Universal Set— reference set for the complement operation

Empty Set— identity element for union, annihilator for intersection

See All Set Theory Definitions →

Union

The union of two sets

A

and

B

is the set of elements that belong to

A

B

or both:

A \cup B = \{x : x \in A \text{ or } x \in B\}

The word "or" here is inclusive — elements belonging to both sets are included. If

A = \{1, 2, 3\}

and

B = \{3, 4, 5\}

, then:

A \cup B = \{1, 2, 3, 4, 5\}

The element

3

appears in both sets but is listed only once in the union.

Union is commutative and associative:

A \cup B = B \cup A

(A \cup B) \cup C = A \cup (B \cup C)

For multiple sets, the union extends naturally:

\bigcup_{i=1}^{n} A_i = A_1 \cup A_2 \cup \cdots \cup A_n

This contains every element that appears in at least one of the sets

A_1, A_2, \ldots, A_n

Intersection

The intersection of two sets

A

and

B

is the set of elements that belong to both

A

and

B

A \cap B = \{x : x \in A \text{ and } x \in B\}

A = \{1, 2, 3\}

and

B = \{3, 4, 5\}

, then:

A \cap B = \{3\}

When two sets have no elements in common, their intersection is the empty set, and the sets are called disjoint:

A \cap B = \emptyset

Intersection is commutative and associative:

A \cap B = B \cap A

(A \cap B) \cap C = A \cap (B \cap C)

For multiple sets:

\bigcap_{i=1}^{n} A_i = A_1 \cap A_2 \cap \cdots \cap A_n

This contains only elements that appear in every one of the sets

A_1, A_2, \ldots, A_n

Complement

The complement of a set

A

consists of all elements not in

A

. This operation requires a universal set

U

that contains all elements under consideration:

A^c = \{x \in U : x \notin A\}

Alternative notations include

A'

and

\overline{A}

.

If the universal set is

U = \{1, 2, 3, 4, 5\}

and

A = \{1, 2\}

, then:

A^c = \{3, 4, 5\}

The complement depends entirely on the choice of universal set. The same set

A

has different complements relative to different universal sets.

Taking the complement twice returns the original set:

(A^c)^c = A

Two additional facts hold for any set

A

A \cup A^c = U

A \cap A^c = \emptyset

Every element either belongs to

A

or to its complement, but not to both.

Set Difference

The set difference

A \setminus B

(also written

A - B

) contains all elements that belong to

A

but not to

B

A \setminus B = \{x : x \in A \text{ and } x \notin B\}

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

and

B = \{3, 4, 5\}

, then:

A \setminus B = \{1, 2\}

B \setminus A = \{5\}

Unlike union and intersection, set difference is not commutative — the order of

A

and

B

matters.

Set difference relates to complement: when

B \subseteq U

, the complement

B^c

equals

U \setminus B

. More generally:

A \setminus B = A \cap B^c

This expresses difference as an intersection with a complement.

Symmetric Difference

The symmetric difference of

A

and

B

contains elements that belong to exactly one of the two sets:

A \triangle B = (A \setminus B) \cup (B \setminus A)

An equivalent definition uses union and intersection:

A \triangle B = (A \cup B) \setminus (A \cap B)

A = \{1, 2, 3\}

and

B = \{3, 4, 5\}

, then:

A \triangle B = \{1, 2, 4, 5\}

The element

3

belongs to both sets, so it is excluded from the symmetric difference.

Symmetric difference is commutative and associative:

A \triangle B = B \triangle A

(A \triangle B) \triangle C = A \triangle (B \triangle C)

The symmetric difference with the empty set leaves a set unchanged:

A \triangle \emptyset = A

. The symmetric difference of a set with itself is empty:

A \triangle A = \emptyset

Summary: All Five Operations at a Glance

The five operations introduced above share a uniform structure: each consumes one or two sets and returns a new set, defined by a precise membership condition. The capstone table below collects them all in one place — every operation with its notation, its membership rule, the result on a single worked example (

A = \{1, 2, 3\}

B = \{3, 4, 5\}

U = \{1, 2, 3, 4, 5\}

), and its key algebraic properties.

Operation	Notation	Elements that...	Result on A = {1, 2, 3}, B = {3, 4, 5}, U = {1, ..., 5}	Key properties
Union	A ∪ B	belong to A or B (inclusive or)	{1, 2, 3, 4, 5}	commutative, associative; identity = ∅
Intersection	A ∩ B	belong to both A and B	{3}	commutative, associative; identity = U; empty if A, B disjoint
Complement	A^c (also A', A, or U \ A)	are in U but not in A (unary operation)	A^c = {4, 5}	(A^c)^c = A; A ∪ A^c = U; A ∩ A^c = ∅
Set difference	A \ B (also A − B)	are in A but not in B	A \ B = {1, 2}; B \ A = {4, 5}	NOT commutative; A \ B = A ∩ B^c
Symmetric difference	A △ B	are in exactly one of A or B	{1, 2, 4, 5}	commutative, associative; identity = ∅; A △ A = ∅