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

Venn Diagrams for Two and Three Sets

Updated 10 October 2026 · Fact-checked

A Venn diagram shows sets as overlapping circles inside a rectangle that stands for the universal set. Each region stands for a group of elements. To solve a question, draw the circles, fill the innermost overlap first, work outwards, then shade or count the region the question asks for.

Understand Venn Diagrams

A set is a collection of well-defined objects. A Venn diagram is a picture of sets. The rectangle is the universal set U, which holds everything under discussion. Each circle inside it is one set.

If two circles overlap, the overlapping part holds elements common to both sets. This is the intersection, A ∩ B. Everything covered by either circle, including the overlap, is the union, A ∪ B. If two sets have nothing in common (disjoint sets), draw the circles apart.

Two overlapping circles split U into four regions: only A, only B, both A and B, and neither. Three overlapping circles split U into eight regions: only A, only B, only C, A and B only, B and C only, A and C only, all three, and none. Drawing all three circles overlapping each other, with a common centre, gives you all eight.

The difference A − B means elements in A that are not in B. It is the crescent of circle A outside circle B. The complement A′ means everything in U outside A, so it is the area of the rectangle outside the circle. If A is a subset of B, draw circle A fully inside circle B.

Key formulas to remember

Union of two sets: n(A ∪ B) = n(A) + n(B) − n(A ∩ B); Subtract the overlap once because it is counted twice when you add n(A) and n(B).
Difference of sets: n(A − B) = n(A) − n(A ∩ B); A − B is only A, the part of A outside B. It is not the same as B − A.
Complement: n(A′) = n(U) − n(A); A′ is everything in the universal set that is not in A.
Union of 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 the pairs, add back the triple overlap.
Neither set: n(neither A nor B) = n(U) − n(A ∪ B); This is the region outside both circles, which is (A ∪ B)′.

How to solve Venn Diagrams questions

Use the same routine for any Venn diagram question, whether you must shade a region or find a number.

1Draw a rectangle for U and the right number of circles. Make all circles overlap if no relation is given.
2Mark any given relation: disjoint circles apart, a subset circle inside the larger one.
3Write the innermost overlap first. For three sets this is A ∩ B ∩ C. For two sets it is A ∩ B.
4Work outwards. Pair overlaps only = pair total − triple overlap. Then single-set only = set total − all overlaps it touches.
5Find the region outside all circles by subtracting the total in circles from n(U).
6Read off the region the question asks for. For shading questions, identify the region in words first, such as 'in A but not in B'.
7Check that all regions add up to n(U).

Quickest way: Fill from the centre and check the total

When to use it: Use this for any word problem that gives counts of people or items in two or three groups.

Skip the formula for three sets. Draw the diagram and fill regions directly.
Start with the triple overlap, then the 'exactly two' regions by subtracting it from each pair total.
Get the 'only one' regions by subtracting from each set total.
Add every region. Anything left from the grand total goes outside the circles.
For shading, translate symbols into words: ∪ is 'or', ∩ is 'and', − is 'but not', ′ is 'not'.

Common mistakes in Venn Diagrams

Treating 'n(A ∩ B) = 12' as the number of elements only in both and then subtracting it again from the overlap region.
Students confuse the given pair total with the 'exactly two' region in a three-set diagram.
Fix: In a three-set diagram, the pair total includes the centre. Subtract n(A ∩ B ∩ C) to get the part of the pair region without C.
Writing the full set total in the 'only A' region.
The number n(A) is given, so it looks natural to put it in circle A directly.
Fix: n(A) is the whole circle, with all overlaps. Subtract the overlaps before writing the 'only A' number.
Shading the wrong region for A − B and shading B − A.
The order of the sets is overlooked.
Fix: A − B means in A but not in B. Shade the part of circle A outside B, leaving the overlap unshaded.
Shading the whole circle for A′.
The complement sign is mistaken for the set itself.
Fix: A′ is the rectangle outside circle A. Shade everything in U except A.
Forgetting the region outside all circles.
Students stop once the circles are filled.
Fix: Compute n(U) minus the total inside the circles. This gives 'neither', and it is often the answer sought.
Drawing disjoint circles when no relation is stated.
Students assume sets with different names have nothing in common.
Fix: Draw overlapping circles by default. If the data show no common elements, the overlap simply becomes 0.

Worked examples

Example 1

In a class of 60 students, 35 like cricket, 28 like football and 10 like both. How many like neither game?

Show the solution

Let C = cricket and F = football. n(U) = 60, n(C) = 35, n(F) = 28, n(C ∩ F) = 10.
Only cricket = 35 − 10 = 25.
Only football = 28 − 10 = 18.
Both = 10.
Students in at least one game = 25 + 18 + 10 = 53. Same as 35 + 28 − 10 = 53.
Neither = 60 − 53 = 7.

Answer: 7 students like neither game.

Example 2

In a survey of 100 customers of a shop in Pune, 50 buy tea (T), 40 buy coffee (C) and 30 buy milk (M). 15 buy T and C, 10 buy C and M, 12 buy T and M, and 5 buy all three. How many buy exactly one of the three items?

Show the solution

Triple overlap = 5.
T and C only = 15 − 5 = 10. C and M only = 10 − 5 = 5. T and M only = 12 − 5 = 7.
Only T = 50 − 10 − 7 − 5 = 28.
Only C = 40 − 10 − 5 − 5 = 20.
Only M = 30 − 5 − 7 − 5 = 13.
Exactly one = 28 + 20 + 13 = 61.
Check: total inside circles = 61 + (10 + 5 + 7) + 5 = 88, so 12 buy none, and 88 + 12 = 100.

Answer: 61 customers buy exactly one item.

Exam tips

Most questions are MCQs asking for a count or the correct shaded region. Sketch a quick diagram even for a two-set problem; it prevents slips.
For 'which region is shaded' options, put the symbols into words and test one region at a time.
In three-set questions, ask whether the given pair totals include the triple overlap. Usually they do.
Always compute the 'neither' region when n(U) is given. Wrong options are often built from forgetting it.
Check that all regions add up to n(U) before marking. There is no negative marking, so always attempt every question.

Practice questions from Set Theory, including Venn Diagram

Venn Diagrams in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

CA FoundationSet Operations and Venn Diagrams

Venn Diagrams: frequently asked questions

How do I draw a Venn diagram for three sets?

Draw a rectangle for U and three circles that each overlap the other two, with a common area in the middle. This gives eight regions, including the one outside all circles. Fill the centre first and work outwards.

How do I shade A − B in a Venn diagram?

Shade the part of circle A that lies outside circle B. The overlap A ∩ B stays unshaded. Remember that A − B and B − A are different regions.

What is the difference between A ∪ B and A ∩ B in a Venn diagram?

A ∪ B is the area covered by either circle, including the overlap. A ∩ B is only the overlapping part. Think 'or' for union and 'and' for intersection.

Do I need to memorise the three-set formula?

Yes, it is a useful check, but the diagram method is safer under time pressure. Filling regions from the centre usually avoids sign errors.