Venn Diagrams and Sets

What is a set?

A set is simply a collection of things. Those things are called the elements (or members) of the set. We list the elements inside curly brackets { }, separated by commas. For example:

• The set of even numbers under 10: {2, 4, 6, 8}
• The set of vowels: {a, e, i, o, u}
• The set of days starting with "S": {Saturday, Sunday}

Sets are usually given capital-letter names like A or B. So we might write A = {2, 4, 6, 8}. The order does not matter, and we never repeat an element — each one is listed just once. Sorting things into groups like this is a natural extension of factors and multiples, where numbers are grouped by what divides them.

Drawing a Venn diagram

A Venn diagram shows sets as circles. Each circle holds the elements of one set. When two sets share some elements, their circles overlap, and the shared elements go in the overlapping part in the middle.

Imagine two sets:

• A = pets that are furry = {cat, dog, rabbit, hamster}
• B = pets that can be walked on a lead = {dog, rabbit}

We draw two overlapping circles. A dog is furry and can be walked, so it goes in the overlap. A cat is only furry, so it sits in the part of circle A outside the overlap. This single picture shows, at a glance, what the two groups share and what is unique to each.

Intersection: what is in BOTH

The intersection of A and B is the set of elements that are in both sets. We write it with the symbol ∩ (it looks like an "n" for and):

A ∩ B = the elements in A AND in B

It is the overlap in the middle of the Venn diagram.

Worked example 1: A = {1, 2, 3, 4} and B = {3, 4, 5, 6}. Find A ∩ B.

1. Look for numbers that appear in both lists.
2. 3 is in both. 4 is in both. The others appear in only one set.
3. So A ∩ B = {3, 4}.

Union: what is in EITHER

The union of A and B is everything that is in either set — you combine all the elements together, listing each only once. The symbol is ∪:

A ∪ B = the elements in A OR in B (or both)

It is the whole area covered by both circles.

Worked example 2: Using A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, find A ∪ B.

1. Write down every element from both sets.
2. Combine: 1, 2, 3, 4, 5, 6 — but do not repeat the 3 and 4 that appear twice.
3. So A ∪ B = {1, 2, 3, 4, 5, 6}.

A handy way to remember the symbols: ∪ looks like a cup that holds everything (union), while ∩ is the smaller cap in the middle (intersection).

The universal set and the complement

Sometimes we fix the full collection of things we are talking about. This is the universal set, written with the symbol ξ (or U). It is drawn as a rectangle around the circles. Anything not inside a circle still sits inside this rectangle.

The complement of set A means everything in the universal set that is NOT in A. It is written A′ (A "prime").

Worked example 3: The universal set is the numbers 1 to 6: ξ = {1, 2, 3, 4, 5, 6}. Let A = {2, 4, 6}. Find the complement A′.

1. The complement is everything in ξ that is not in A.
2. Remove 2, 4 and 6 from the universal set.
3. So A′ = {1, 3, 5} (the odd numbers).

On the diagram, A′ is everything outside circle A but still inside the rectangle.

A worked Venn diagram with numbers

A class of students was asked whether they play football (F) or basketball (B):

• 12 students play football
• 9 students play basketball
• 5 students play both sports
• 4 students play neither

Worked example 4: Fill in the Venn diagram and find the total class size.

1. Start with the overlap: 5 play both, so put 5 in the middle.
2. Football only: 12 play football, but 5 of those also play basketball, so 12 − 5 = 7 go in the football-only part.
3. Basketball only: 9 − 5 = 4 go in the basketball-only part.
4. Neither: 4 go outside both circles (but inside the rectangle).
5. Total = 7 + 5 + 4 + 4 = 20 students.

Notice why we subtract: if we just did 12 + 9 = 21, we would count the 5 "both" students twice. Venn diagrams stop that double-counting.

A symbols reference table

Symbol	Name	Meaning
{ }	Set brackets	Holds the elements of a set
∈	Element of	"is in", e.g. 3 ∈ {1, 2, 3}
∩	Intersection	In both sets (the overlap)
∪	Union	In either set (everything)
A′	Complement	Everything not in A
ξ	Universal set	The full collection (the rectangle)

Where Venn diagrams are used

Venn diagrams turn up far beyond maths class. Computer searches use the same idea: typing "cats AND dogs" finds the intersection, while "cats OR dogs" finds the union. Scientists compare groups — which animals have fur, which lay eggs, which do both. Probability uses Venn diagrams to work out the chance of overlapping events, which links closely to probability basics. Even sorting your own life — homework due today and tomorrow — is set thinking.

Why they help: A Venn diagram makes invisible relationships visible. Instead of juggling lists in your head, you can see what two groups share, what is unique, and what is left out — all in one picture.

Practice activity

Use A = {2, 3, 5, 7} and B = {2, 4, 6, 8}, with universal set ξ = {1, 2, 3, 4, 5, 6, 7, 8}.

1. Find A ∩ B.
2. Find A ∪ B.
3. Find A′ (the complement of A).
4. In a survey, 10 people like tea, 8 like coffee, 3 like both, and 2 like neither. How many people were surveyed?

Answers:

1. The only number in both sets is 2, so A ∩ B = {2}.
2. Combine without repeats: A ∪ B = {2, 3, 4, 5, 6, 7, 8}.
3. Everything in ξ not in A: A′ = {1, 4, 6, 8}.
4. Tea only = 10 − 3 = 7; coffee only = 8 − 3 = 5; both = 3; neither = 2. Total = 7 + 5 + 3 + 2 = 17 people.

Summary

A set is a collection of elements written in curly brackets, and a Venn diagram pictures sets as overlapping circles. The intersection (A ∩ B) is what is in both sets — the overlap; the union (A ∪ B) is everything in either set, counted once. The universal set is the full collection (the rectangle), and the complement (A′) is everything outside a set. Filling in a Venn diagram from the overlap outward stops you from double-counting — which is why it is such a clear way to compare and sort information.