Sets and Venn Diagrams

What is a Set?

A set is a well-defined collection of distinct objects, called elements or members. Sets are written with curly braces, and each element is listed once.

Examples:

• A = {2, 4, 6, 8} — even single-digit numbers
• B = {cat, dog, rabbit} — some common pets
• C = {} — the empty set (written ∅), which contains no elements

The order of elements inside a set does not matter: {1, 2, 3} is the same set as {3, 1, 2}.

Set Notation

Symbol	Meaning	Example
∈	is an element of	4 ∈ {2, 4, 6}
∉	is not an element of	5 ∉ {2, 4, 6}
n(A)	number of elements in A	n({2, 4, 6}) = 3
∅ or {}	empty set	no elements
ξ	universal set	all elements under consideration
⊆	is a subset of	{2, 4} ⊆ {2, 4, 6}

A is a subset of B (written A ⊆ B) if every element of A is also in B. Every set is a subset of itself, and the empty set is a subset of every set.

Union and Intersection

Union (A ∪ B) — everything in A or B (or both). Think of it as combining the two sets.

Intersection (A ∩ B) — everything in both A and B. Think of it as the overlap.

Example: A = {1, 2, 3, 4} and B = {3, 4, 5, 6}

• A ∪ B = {1, 2, 3, 4, 5, 6} — all elements from either set, listed once
• A ∩ B = {3, 4} — only the elements appearing in both

When two sets share no elements, they are called disjoint and A ∩ B = ∅.

Complement

The complement of A, written A′, contains every element in the universal set ξ that is not in A.

Example: ξ = {1, 2, 3, 4, 5, 6, 7, 8}, A = {2, 4, 6, 8} A′ = {1, 3, 5, 7}

Notice that A and A′ together make up the entire universal set, and they share no elements: A ∩ A′ = ∅.

Venn Diagrams

A Venn diagram uses overlapping circles inside a rectangle to show the relationships between sets. The rectangle represents the universal set ξ.

ξ
┌─────────────────────────────────┐
│   ┌──────────┐  ┌──────────┐   │
│   │  A only  │∩ │  B only  │   │
│   │  {1, 2}  │  │  {5, 6}  │   │
│   │      {3, 4}            │   │
│   └──────────┘  └──────────┘   │
└─────────────────────────────────┘

• Left region (A only): elements in A but not B → {1, 2}
• Overlapping region: elements in both A and B → A ∩ B = {3, 4}
• Right region (B only): elements in B but not A → {5, 6}
• Outside both circles: elements in ξ but in neither A nor B

Venn diagrams with three sets work the same way — three overlapping circles create seven distinct regions (each subset combination).

The Counting Formula

When two sets overlap, simply adding their sizes double-counts the shared elements. The correct formula is:

n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

Example: In a class of 30 students, 18 study French, 15 study Spanish, and 8 study both.

n(French ∪ Spanish) = 18 + 15 − 8 = 25 students study at least one language.

That means 30 − 25 = 5 students study neither.

This formula appears frequently in exam problems where you are given totals and need to find an unknown region.

Solving Venn Diagram Problems

A common exam style: you are given totals for each region and asked to find a missing value.

Example: ξ has 40 elements. n(A) = 22, n(B) = 18, n(A ∩ B) = 10.

Step 1 — Fill in the intersection: A ∩ B = 10. Step 2 — A only: 22 − 10 = 12. B only: 18 − 10 = 8. Step 3 — Elements in A ∪ B: 12 + 10 + 8 = 30. Step 4 — Neither: 40 − 30 = 10.

Draw the diagram, fill each region in order, and use the total to find anything remaining.

Useful Facts to Remember

• n(A) + n(A′) = n(ξ) — a set and its complement together fill the universal set
• A ∩ A = A and A ∪ A = A — a set intersected or united with itself is just itself
• A ∩ ∅ = ∅ and A ∪ ∅ = A — the empty set contributes nothing to a union and absorbs the intersection
• (A′)′ = A — the complement of a complement returns the original set

Common Mistakes

• Listing elements twice in a union: each element appears once, no matter how many sets it belongs to.
• Confusing ∪ and ∩: ∪ (union) looks like a cup that holds all the water; ∩ (intersection) looks like a cap covering only the middle part.
• Forgetting to subtract the overlap: when using n(A ∪ B) = n(A) + n(B) − n(A ∩ B), the subtraction is essential whenever A and B share elements.
• Placing elements in the wrong Venn region: elements in A ∩ B go in the overlapping middle section, not in one circle alone.