Operations on Sets — Union, Intersection, Complement

Operations on Sets — Union, Intersection, Complement

Algebra

TL;DR

Operations on sets are the rules that combine or compare sets — union ∪\cup∪, intersection ∩\cap∩, difference −-−, and complement ′'′ — each producing a new set. This article defines all four with their symbols and Venn diagrams, lists the key properties (commutative, associative, distributive, De Morgan's laws), gives the cardinality formula, and works through common mistakes.

BT
Last updated on June 29, 2026 8 min read

What Are Operations On Sets?

Operations on sets are rules that take one or more sets and return a new set. The four core operations are union, intersection, difference, and complement. A set is a well-defined collection of objects, and these operations let us combine and compare those collections without listing every element by hand.

Throughout, take a universal set U — the full collection everything is drawn from — and two sets inside it:

U={1, 2, 3, 4, 5, 6, 7, 8}
A={1, 2, 3, 4}
B={3, 4, 5, 6}

These three sets run through every operation below, so you can watch one example deepen.

Union — combining everything

The union A∪B is the set of elements in A, in B, or in both. The symbol ∪ reads "or".

A∪B={1, 2, 3, 4, 5, 6}

Shared elements are listed once — a set never repeats a member.

Intersection — keeping only the shared

The intersection A∩B is the set of elements in both A and B. The symbol ∩ reads "and".

A∩B={3, 4}

When two sets share nothing, the intersection is the empty set, and the sets are called disjoint.

Difference — what one has that the other lacks

The difference A−B (also written A∖B) is the set of elements in A but not in B. Order matters here.

A−B={1, 2}, B−A={5, 6}

A−B and B−A are different sets — a clue that difference is not commutative.

Complement — everything outside

The complement A′ (also written A^c) is the set of elements in the universal set U but not in A.

A'={5, 6, 7, 8}

Complement only makes sense once a universal set is fixed; without U, "everything outside A" has no boundary.

Symmetric difference — in one or the other, not both

The symmetric difference A△B collects elements in exactly one of the two sets.

A△B={1, 2, 5, 6}

It is the union minus the intersection — useful when "either but not both" is the real question.

Properties Of Operations On Sets

The operations obey laws that mirror, but do not copy, the laws of arithmetic. Keep them straight and most set problems collapse to one line.

The cardinality of a union ties three counts together:

n(A∪B)=n(A)+n(B)−n(A∩B)

You subtract the intersection because adding n(A) and n(B) counts the shared elements twice.

Examples of Operations on Sets

Example 1

With A={2, 4, 6} and B={4, 6, 8}, find A∪B and A∩B.

A∪B={2, 4, 6, 8}
A∩B={4, 6}

Final answer: A∪B={2, 4, 6, 8}, A∩B={4, 6}.

Example 2

Given U={1, 2, 3, 4, 5}, A={1, 2, 3}, B={2, 3, 4}, find (A∪B)′.

Direct: (A∪B)′={5}.

Example 3

With A={a, b, c, d} and B={c, d, e}, find A−B and B−A.

A−B={a, b}
B−A={e}

Final answer: A−B={a, b}, B−A={e}.

Example 4

Verify the distributive law for A={1, 2}, B={2, 3}, C={1, 3}.

Left side: A∩(B∪C)={1, 2}, Right side: (A∩B)∪(A∩C)={1, 2}.

Example 5

In a group of 40 people, 25 like tea and 20 like coffee; 10 like both. How many like at least one?

n(T∪C)=25+20−10=35.

Example 6

How many like exactly one of the two drinks?

That is the symmetric difference. n(T△C)=(n(T)−n(T∩C))+(n(C)−n(T∩C))=25.

Where Operations On Sets Go Sideways

Mistake 1

Misapplying De Morgan's laws. The correct way: (A∩B)′=A′∪B′.

Mistake 2

Treating set difference as commutative.

Mistake 3

Forgetting the universal set for complement. Complement is always relative to a fixed universal set U.

Conclusion