Venn Diagrams

Venn Diagrams

Illustrations of set operations as shown below.

Two Venn diagrams: Intersection (A∩B, overlapping region shaded) and Union (A∪B, both circles shaded).

Two Venn diagrams showing set subtraction: A−B (left set shaded) and B−A (right set shaded), each with overlapping circles A...

Two Venn diagrams: complement A^C shown as shaded region outside oval A; subset showing oval A inside oval B, where B⊂A and A⊃B.

Key Formula

|A \cup B| = |A| + |B| - |A \cap B|

Where:

$|A \cup B|$ = Number of elements in the union of sets A and B (everything in A or B or both)
$|A|$ = Number of elements in set A
$|B|$ = Number of elements in set B
$|A \cap B|$ = Number of elements in the intersection of A and B (elements in both A and B)

Worked Example

Problem: In a class of 40 students, 25 play soccer, 18 play basketball, and 10 play both. Use a Venn diagram to find how many students play neither sport.

Step 1: Identify the given information. Let A = soccer players and B = basketball players.

|A| = 25, \quad |B| = 18, \quad |A \cap B| = 10, \quad |\text{Universal}| = 40

Step 2: Place the overlap first. The intersection region (both sports) contains 10 students.

|A \cap B| = 10

Step 3: Find the 'soccer only' region by subtracting the overlap from the total soccer count. Similarly find 'basketball only'.

\text{Soccer only} = 25 - 10 = 15, \quad \text{Basketball only} = 18 - 10 = 8

Step 4: Apply the inclusion–exclusion formula to find the total number who play at least one sport.

|A \cup B| = 25 + 18 - 10 = 33

Step 5: Subtract from the universal set to find how many play neither sport.

\text{Neither} = 40 - 33 = 7

Answer: 7 students play neither soccer nor basketball. The Venn diagram has four regions: soccer only (15), both (10), basketball only (8), and neither (7).

Another Example

This example focuses on listing specific elements in each region of the Venn diagram and performing multiple set operations, rather than counting with the inclusion–exclusion formula.

Problem: Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 2, 3, 4, 5}, and B = {3, 4, 5, 6, 7}. Use a Venn diagram to find A ∩ B, A ∪ B, A \ B, and A'.

Step 1: Find the intersection — elements that belong to both A and B.

A \cap B = \{3, 4, 5\}

Step 2: Find the union — all elements that belong to A or B or both.

A \cup B = \{1, 2, 3, 4, 5, 6, 7\}

Step 3: Find the set subtraction A \ B — elements in A that are not in B.

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

Step 4: Find the complement of A — elements in the universal set that are not in A.

A' = U \setminus A = \{6, 7, 8, 9, 10\}

Answer: A ∩ B = {3, 4, 5}, A ∪ B = {1, 2, 3, 4, 5, 6, 7}, A \ B = {1, 2}, and A' = {6, 7, 8, 9, 10}. On the Venn diagram, 8, 9, and 10 sit outside both circles.

Frequently Asked Questions

What is the difference between union and intersection in a Venn diagram?

The union (A ∪ B) is the entire shaded area covered by both circles — every element in A, in B, or in both. The intersection (A ∩ B) is only the overlapping region where both circles meet, containing elements that belong to A and B simultaneously. On a Venn diagram, union is a larger region and intersection is a smaller one (or empty if the sets are disjoint).

How do you read a three-circle Venn diagram?

A three-circle Venn diagram has 8 distinct regions: three 'only' sections (A only, B only, C only), three pairwise overlaps (A∩B only, A∩C only, B∩C only), the central overlap of all three (A∩B∩C), and the area outside all circles. You fill in values starting from the innermost region (all three) and work outward, subtracting as you go.

Why do you subtract the intersection in the inclusion–exclusion formula?

When you add |A| + |B|, every element in the overlap A ∩ B gets counted twice — once as part of A and once as part of B. Subtracting |A ∩ B| corrects this double-counting so each element is counted exactly once. This is the core idea behind the inclusion–exclusion principle.

Venn Diagram vs. Carroll Diagram (Two-Way Table)

	Venn Diagram	Carroll Diagram (Two-Way Table)
Visual format	Overlapping circles inside a rectangle	Grid of rows and columns
Best for	Showing intersections, unions, and complements of 2–3 sets	Organizing data by two categorical attributes (yes/no for each)
Handles more than 3 sets?	Becomes complex and hard to read beyond 3 circles	Scales to larger tables more easily
Shows overlap visually	Yes — overlapping regions are immediately visible	No — relationships are read from cell positions

Why It Matters

Venn diagrams appear throughout probability, statistics, and logic courses. In probability, they help you visualize events and apply the addition rule P(A ∪ B) = P(A) + P(B) − P(A ∩ B). They are also a standard tool in discrete mathematics and computer science for reasoning about database queries, Boolean logic, and survey analysis problems.

Common Mistakes

Mistake: Forgetting to subtract the intersection when counting the union, which leads to double-counting the overlapping elements.

Correction: Always use |A ∪ B| = |A| + |B| − |A ∩ B|. Start by placing the intersection count in the overlap region first, then compute the remaining portions of each circle.

Mistake: Ignoring the region outside all circles (the complement of the union) and assuming every element belongs to at least one set.

Correction: Remember the rectangle represents the universal set. After filling in all circle regions, subtract their total from the universal set to find the 'neither' region: |Neither| = |U| − |A ∪ B|.

Related Terms

Set — The fundamental objects displayed in a Venn diagram
Intersection — The overlapping region of two or more circles
Union — The total area covered by all circles combined
Set Subtraction — Region of one circle excluding the overlap
Complement of a Set — Everything in the rectangle outside a circle
Subset — One circle entirely inside another
Superset — The outer circle that fully contains another