Fundamentals of Business Mathematics and Statistics · Set Theory, including Venn Diagram

Set Operations: Union, Intersection, Difference and Complement

Updated 10 October 2026 · Fact-checked

Set operations combine or compare sets to form new sets. Union (A ∪ B) collects all elements in either set. Intersection (A ∩ B) keeps common elements. Difference (A − B) keeps elements of A not in B. Complement (A′) holds elements of the universal set not in A. List elements carefully and apply the definition.

Understand Set Operations: Union, Intersection, Difference, Complement

A set is a well-defined collection of objects. Set operations tell you how to build a new set from sets you already have. Every question in this topic is really about asking, for each element: is it in or out?

The universal set (U) is the big set that contains everything under discussion. All other sets are subsets of U. You need U to find a complement.

Union A ∪ B is the set of elements that are in A, or in B, or in both. Think of it as "merge and write each element once". Intersection A ∩ B is the set of elements that are in both A and B. Think of it as "what is common".

Difference A − B is the set of elements that are in A but not in B. Order matters: A − B and B − A are usually different. The symmetric difference A Δ B holds elements that are in exactly one of the two sets, so it is (A − B) ∪ (B − A). The complement A′ is U − A: everything in U that is not in A.

Two sets are disjoint if they have no common element, that is A ∩ B = ∅. For disjoint sets, the union is just the two sets put together, and A − B = A.

Key formulas to remember

Union: A ∪ B = {x : x ∈ A or x ∈ B}; "Or" is inclusive. Write common elements only once.
Intersection: A ∩ B = {x : x ∈ A and x ∈ B}; If there is no common element, the answer is ∅.
Difference: A − B = {x : x ∈ A and x ∉ B}; A − B = A ∩ B′. Not the same as B − A in general.
Symmetric difference: A Δ B = (A − B) ∪ (B − A) = (A ∪ B) − (A ∩ B); Elements in exactly one of the two sets.
Complement: A′ = U − A; (A′)′ = A; U′ = ∅; ∅′ = U; Always depends on the universal set U.
Disjoint sets: A ∩ B = ∅, so n(A ∪ B) = n(A) + n(B); Disjoint means no common element.
Basic properties: A ∪ A = A; A ∩ A = A; A ∪ ∅ = A; A ∩ ∅ = ∅; A ∪ A′ = U; A ∩ A′ = ∅; Union and intersection are commutative: A ∪ B = B ∪ A, A ∩ B = B ∩ A.
Subset links: If A ⊆ B then A ∪ B = B, A ∩ B = A and A − B = ∅; Useful for quick checks.

How to solve Set Operations: Union, Intersection, Difference, Complement questions

Use this method for any question that asks you to find a set after one or more operations.

1Write down U, A and B as lists of elements. If sets are given by a rule, convert them to lists first.
2Note the operation symbol and read it in words: union is "or", intersection is "and", difference is "in first, not in second".
3If the expression has brackets or a complement, work out the inside first. Find any complement using U.
4Go through the elements one by one and decide in or out. Do not rely on memory.
5Write the final answer in braces, each element once, in increasing order.
6Check the answer: it must be a subset of U, and no element should repeat.
7If a statement is to be checked as true or false, test it with the given sets, not with a general belief.

Quickest way: Element-tick method

When to use it: Use it for MCQs where sets are small lists and you must pick the correct resulting set.

Write U, A and B in a row.
Complement first: cross out A's elements from U to get A′ in seconds.
For intersection, pick only elements you see in both lists; the answer is often short, so check options of the right size first.
For difference A − B, strike out from A anything that appears in B.
Eliminate options that contain an element outside U or an element repeated.
For symmetric difference, take union and remove the common elements.

Common mistakes in Set Operations: Union, Intersection, Difference, Complement

Treating A − B and B − A as the same set.
Students think of difference like subtraction of numbers that can be swapped, or forget that order matters.
Fix: Always ask "in the first, not in the second". Check which set comes first.
Writing common elements twice in a union.
Students join the two lists without checking repeats.
Fix: After joining, scan and keep each element once. For example {1, 2, 3} ∪ {3, 4} = {1, 2, 3, 4}.
Finding a complement without using the universal set.
The complement seems to be "everything else", so students pick elements at random.
Fix: Take U and remove the elements of A. If U is not stated, the question cannot be answered without it.
Confusing symmetric difference with union or intersection.
The symbol Δ is less familiar.
Fix: Remember it as "exactly one". Common elements are removed, so A Δ B = (A ∪ B) − (A ∩ B).
Saying the intersection of disjoint sets is {0} or {∅}.
Students mix up "empty" and "zero" or "set containing the empty set".
Fix: The intersection of disjoint sets is ∅, a set with no elements.
Doing operations left to right when brackets or complements are present.
Rushing under time pressure.
Fix: Do brackets and complements first, then the remaining operation.

Worked examples

Example 1

Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 2, 3, 4, 5, 6} and B = {4, 5, 6, 7, 8}. Find A Δ B and (A ∪ B)′.

Show the solution

A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8}.
A ∩ B = {4, 5, 6}.
A Δ B = (A ∪ B) − (A ∩ B) = {1, 2, 3, 7, 8}.
Check with the other form: A − B = {1, 2, 3} and B − A = {7, 8}. Their union is {1, 2, 3, 7, 8}. It matches.
(A ∪ B)′ = U − {1, 2, 3, 4, 5, 6, 7, 8} = {9, 10}.

Answer: A Δ B = {1, 2, 3, 7, 8} and (A ∪ B)′ = {9, 10}.

Example 2

If U = {1, 2, 3, 4, 5, 6, 7, 8}, A = {2, 4, 6, 8} and B = {1, 2, 3, 4}, which one is the set A′ ∩ B? Options: (a) {1, 3} (b) {2, 4} (c) {5, 7} (d) {1, 3, 5, 7}

Show the solution

Find A′ = U − A = {1, 3, 5, 7}.
A′ ∩ B means elements common to A′ and B.
B = {1, 2, 3, 4}. Elements of B that are in A′ are 1 and 3.
So A′ ∩ B = {1, 3}.
Elimination check: option (b) is A ∩ B, option (c) is A′ − B, and option (d) is A′ itself, so each fails.

Answer: Option (a): {1, 3}

Exam tips

Most questions give small lists. Write U, A and B clearly first; it takes ten seconds and prevents most errors.
Always compute the complement before anything else when it appears inside a bracket.
Use option elimination: any option with an element outside U, or with repeats, is wrong.
Watch the symbol: ∪, ∩, − and Δ look similar in a hurry. Underline it in the question.
For true/false statements about sets, test with the given sets and check the claim for each element.

Practice questions from Set Theory, including Venn Diagram

Set Operations: Union, Intersection, Difference, Complement: frequently asked questions

What is the difference between union and intersection of sets?

Union collects every element that is in at least one of the sets. Intersection keeps only the elements that are in both sets. The union is always at least as large as the intersection.

How do I find the complement of a set?

Take the universal set U and remove all elements of the given set. What remains is the complement. Without U, the complement cannot be found.

What is the symmetric difference of two sets?

It is the set of elements that belong to exactly one of the two sets. You can find it as (A − B) ∪ (B − A), or as (A ∪ B) − (A ∩ B).

Is A − B the same as B − A?

No, not in general. A − B keeps elements of A that are not in B, while B − A keeps elements of B that are not in A. They are equal only when both are empty, that is, when A = B.

What are disjoint sets?

Disjoint sets have no element in common, so their intersection is the empty set. For such sets the number of elements in the union is the sum of the number of elements in each.