Venn Diagram — Union, Intersection, Complement Examples

Venn Diagram — Union, Intersection, Complement Examples

TL;DR

A Venn diagram is a picture of overlapping circles that shows how sets share or separate their elements. This article covers the union, intersection, and complement operations, two-set and three-set diagrams, every region they create, the counting formulas, and step-by-step word problems.

What Is a Venn Diagram?

A Venn diagram is a diagram that uses overlapping closed curves — almost always circles — to represent sets and show the logical relationships between them. Each circle holds the elements of one set; the rectangle around them holds the universal set UUU, every element under discussion.
Where two circles overlap, the elements belong to both sets at once. Where they don't overlap, the elements belong to one set only. The empty space inside the rectangle but outside every circle holds elements that belong to no set in the picture. That is the entire grammar of the diagram, and it powers everything below.
The idea is named for the English logician John Venn, who formalised it in 1880, though the picture of overlapping regions had appeared in logic long before him. A reader can recognise a related tool — the Euler diagram — which drops the overlaps that happen to be empty; a Venn diagram always draws every possible overlap, even ones that turn out to contain nothing.

The Venn Diagram Symbols You Need First

Before the operations make sense, three symbols carry most of the weight. Each describes a relationship between a single element and a set.

The operation symbols — ∪\cup∪ for union, ∩\cap∩ for intersection, and the prime A′ for complement — define the regions we shade. They are the verbs to the nouns above.

Venn Diagrams of Set Operations

Every set operation corresponds to a region you can shade. Once you can read the shading, you can read any Venn diagram. Here are the four core operations on two sets A and B.

Union — A∪B

The union A∪B is the set of elements in A, in B, or in both. On the diagram, you shade everything inside either circle — the two circles together, overlap included.
A∪B={x:x∈A;or;x∈B}

Intersection — A∩B

The intersection A∩B is the set of elements in both A and B. You shade only the lens-shaped overlap.
A∩B={x:x∈A;and;x∈B}
If A and B share no elements, A∩B=∅ — the circles are drawn apart, and the sets are called disjoint.

Complement — A′

The complement A′ (sometimes written Ac) is everything in the universal set U that is not in A. You shade the whole rectangle except circle A.
A′={x:x∈U;and;x∉A}

Difference — A−B

The difference A−B (also A∖B) is the elements in A but not in B. You shade circle A and then erase the overlap. The reverse, B−A, shades B and erases the overlap — the two are not the same.
A close relative is the symmetric difference A△B: everything in exactly one set, which shades both circles but leaves the overlap blank.

How Do You Read a Three-Set Venn Diagram?

This is the question that trips up most students once two sets feel comfortable. Three overlapping circles split the universal set into eight regions, and naming them is the key skill.
The eight regions are:

  1. In A only
  2. In B only
  3. In C only
  4. In A∩B but not C
  5. In B∩C but not A
  6. In A∩C but not B
  7. In A∩B∩C — the centre, in all three
  8. Outside all three — in U but in none
    Every three-set word problem is just these eight regions, filled in. The trick that the strongest solvers use — and that we drill explicitly — is to start from the centre region (all three) and work outward, subtracting as you go, so no element gets counted twice.

How Do You Draw a Venn Diagram?

Drawing one is a fixed routine, and doing it the same way every time is what stops the counting errors later.

  1. Draw the rectangle first — that's the universal set U, the boundary of everything under discussion.
  2. Draw one circle per set inside the rectangle. If the sets share elements, overlap the circles; if they share nothing, keep them apart.
  3. Fill the innermost region first — the deepest overlap (the centre, for three sets) — then work outward to the pair-overlaps, then the single-set regions, then the outside.
  4. Label every region with its count or its elements, and check the parts add up to the whole.

The Counting Formulas

When you only need how many elements, not which ones, the inclusion–exclusion formulas do the work without drawing. For two sets:
n(A∪B)=n(A)+n(B)−n(A∩B)
For three sets:
n(A∪B∪C)=n(A)+n(B)+n(C)−n(A∩B)−n(B∩C)−n(A∩C)+n(A∩B∩C)
Add the singles, subtract every pair-overlap, then add the triple-overlap back — it got subtracted three times by the pairs, so it needs returning once. This add–subtract–add rhythm is the inclusion–exclusion principle, and it generalises to any number of sets.

Examples of Venn Diagrams

The examples build from a single shading up to a full three-set survey. Read them in order; each uses the region-reading skill from the section above.

Example 1

Given A={1,2,3,4} and B={3,4,5,6}, find A∪B and A∩B, and shade each.
The union collects every element that appears in either set, listed once:
A∪B={1,2,3,4,5,6}
The intersection collects only the shared elements:
A∩B={3,4}
Union shades both circles; intersection shades only the overlap holding 3 and 4.

Example 2

A student is told n(A)=24, n(B)=20, and n(A∪B)=40. Find n(A∩B).
Correct. Use the two-set formula and solve for the overlap:
n(A∪B)=n(A)+n(B)−n(A∩B)
40=24+20−n(A∩B)
n(A∩B)=44−40=4
So 4 students do both. The formula refuses to let the overlap be counted twice — that is exactly the protection a Venn diagram gives the eye.

Example 3

Of 50 students, 10 take guitar lessons, 20 take singing, and 4 take both. How many take neither?
First, the number taking at least one:
n(guitar∪singing)=10+20−4=26
Then subtract from the universal set of 50:
n(neither)=50−26=24
Final answer: 24 students take neither.

Example 4

Given U={1,2,…,10}, A={2,4,6,8}, find A′ and A−{4,6}.
The complement is everything in U that is not in A:
A′={1,3,5,7,9,10}
The difference removes only the listed elements from A:
A−{4,6}={2,8}

Example 5

In a class of 90, every student likes at least one of football, cricket, or hockey. 24 like football, 33 like cricket, 38 like hockey; 5 like football and cricket, 8 football and hockey, 10 cricket and hockey; 3 like all three. Verify the total.
n(F∪C∪H)=24+33+38−5−8−10+3=75
That leaves 90−75=15 who like none — but the problem says everyone likes at least one, so the numbers describe a survey where 75 of a larger count are accounted for.

Example 6

A set relationship: if A⊂B, what does A∩B equal, and what does A∪B equal?
When A sits entirely inside B, every element of A is already in B. So their shared part is all of A:
A∩B=A
And their combined part is just the bigger set:
A∪B=B
Final answer: A∩B=A and A∪B=B.

Where Venn Diagrams Show Up in the Real World

Venn diagrams started as a tool of formal logic, but the overlapping-regions idea now runs through fields that have nothing to do with circles on paper.

Tripping Points to Avoid With Venn Diagram

Mistake 1: Counting the overlap twice

Where it slips in: Computing n(A∪B) by adding n(A)+n(B) and stopping.
Don't do this: Report the union as the plain sum of the two set sizes.
The correct way: Subtract the overlap once — n(A∪B)=n(A)+n(B)−n(A∩B).

Mistake 2: Treating A−B and B−A as the same

Where it slips in: Difference problems, where the rusher reads "difference" as symmetric like subtraction of numbers.
Don't do this: Assume A−B=B−A.
The correct way: A−B is "in A, not in B"; B−A is "in B, not in A".

Mistake 3: Filling a three-set diagram from the outside in

Where it slips in: Three-set word problems, where the totals given are the full set sizes, not the "only" regions.
Don't do this: Write 24 in the football-only region just because 24 like football.
The correct way: Start from the centre (all three), then each pair-region, then the singles — subtracting what you've already placed.

The Short Version