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

Cardinality and Applications of Sets with Venn Diagram Word Problems

Updated 10 October 2026 · Fact-checked

Cardinality n(A) is the number of elements in a set. For two sets, n(A∪B) = n(A) + n(B) − n(A∩B). For three sets, add all single sets, subtract all pair overlaps, and add back the triple overlap. For word problems, fill a Venn diagram from the innermost region outwards.

Understand Cardinality and Applications of Sets

The cardinality of a finite set is the count of its distinct elements. We write it as n(A). If A = {2, 4, 6, 8}, then n(A) = 4.

Now think of two groups of people: those who like tea and those who like coffee. If you add the two group sizes, anyone who likes both gets counted twice. So to find how many like at least one drink, you subtract the overlap once. That is the whole idea behind n(A∪B) = n(A) + n(B) − n(A∩B).

With three sets, the same logic applies. Adding all three sets counts people in two sets twice and people in all three thrice. Subtracting the pair overlaps removes too much, because the people in all three get removed three times. So you add n(A∩B∩C) back once.

Word problems are usually survey questions: 100 students, some study one subject, some two, some none. Each region of the Venn diagram is a separate count, and the regions add up to the total. The universal set also includes those in none of the sets, so n(U) = n(A∪B∪C) + n(none).

The skill is translating words into regions. 'Only A' means in A but not in B or C. 'At least one' means the union. 'Exactly two' means the pair overlaps with the triple part removed.

Key formulas to remember

Two sets (union): n(A∪B) = n(A) + n(B) − n(A∩B); Valid for finite sets. If A and B are disjoint, n(A∩B) = 0.
Three sets (union): n(A∪B∪C) = n(A) + n(B) + n(C) − n(A∩B) − n(B∩C) − n(A∩C) + n(A∩B∩C); Add singles, subtract pairs, add the triple.
Only A (two sets): n(A only) = n(A) − n(A∩B); Same as n(A − B).
Only A (three sets): n(only A) = n(A) − n(A∩B) − n(A∩C) + n(A∩B∩C); Remove both overlaps but add back the triple, which was removed twice.
Neither / none: n(none) = n(U) − n(A∪B∪C); Complement of the union. Written n(A′∩B′∩C′).
Exactly two of three: Exactly two = n(A∩B) + n(B∩C) + n(A∩C) − 3·n(A∩B∩C); Each pair overlap includes the triple region, so subtract it three times.

How to solve Cardinality and Applications of Sets questions

Use this method for any survey or Venn diagram word problem. It works for two or three sets.

1Name each set with a letter and write down every given number with its meaning, such as n(A) or n(A∩B).
2Draw a rectangle for the universal set and overlapping circles for the sets.
3Fill the innermost region first. For three sets, that is n(A∩B∩C).
4Fill each pair-only region: n(A∩B) minus the triple, and so on for the other pairs.
5Fill each only-one region: the set total minus everything already placed in that circle.
6Place the 'none' region as n(U) minus the sum of all circle regions.
7Read off the region the question asks for, and check that all regions add up to n(U).

Quickest way: Region-filling from the centre outwards

When to use it: Use it for any three-set survey problem. Use the direct formula only if the question asks for a union and gives all the needed values.

If the question gives n(A∪B∪C) and asks for a missing value such as the triple overlap, put the numbers into the three-set formula and solve for the unknown.
If the question asks for 'only' or 'exactly' regions, skip the formula and fill the diagram.
Do the arithmetic in the diagram, not in your head. Write each region number in the circle.
Before marking an option, check that the regions add up to the total. This catches most slips in under ten seconds.
Remember that there is no negative marking, so if time is short, eliminate options that are larger than the total or negative, then guess.

Common mistakes in Cardinality and Applications of Sets

Writing the pair overlap n(A∩B) in the pair-only region.
n(A∩B) includes the people who are also in C, but students treat it as the part without C.
Fix: Pair-only = n(A∩B) − n(A∩B∩C). Always subtract the triple first.
Forgetting to add back n(A∩B∩C) in the three-set formula.
Students remember 'subtract the overlaps' and stop there.
Fix: Remember the pattern: plus, minus, plus. Singles add, pairs subtract, triple adds.
Ignoring people who belong to none of the sets.
The word 'none' is not in a circle, so it is easy to forget.
Fix: Always check whether the total exceeds the union. The difference is the 'none' region outside the circles.
Confusing 'only A' with 'A'.
Both are about set A, so the words look alike.
Fix: 'A' includes the overlaps. 'Only A' is the part of A outside B and C.
Treating 'at least two' and 'exactly two' as the same.
Students overlook that 'at least two' includes the triple region.
Fix: At least two = exactly two + all three. Choose the correct regions before adding.
Using n(A∪B) = n(A) + n(B) when sets overlap.
This works for disjoint sets, and students use it by habit.
Fix: Check whether the question states a common part. If so, subtract it once.

Worked examples

Example 1

In a class of 60 students, 35 like Mathematics, 28 like Statistics and 10 like both. How many like neither subject?

Show the solution

Let M = Mathematics and S = Statistics. n(M) = 35, n(S) = 28, n(M∩S) = 10.
n(M∪S) = 35 + 28 − 10 = 53.
Neither = n(U) − n(M∪S) = 60 − 53 = 7.

Answer: 7 students like neither subject.

Example 2

In a survey of 100 people, 50 read newspaper A, 40 read B and 30 read C. 15 read A and B, 12 read B and C, 10 read A and C, and 5 read all three. How many read exactly one newspaper?

Show the solution

Triple region: A∩B∩C = 5.
Pair-only regions: A and B only = 15 − 5 = 10; B and C only = 12 − 5 = 7; A and C only = 10 − 5 = 5.
Only A = 50 − 10 − 5 − 5 = 30.
Only B = 40 − 10 − 7 − 5 = 18.
Only C = 30 − 7 − 5 − 5 = 13.
Exactly one = 30 + 18 + 13 = 61.
Check using the union: 50 + 40 + 30 − 15 − 12 − 10 + 5 = 88. Regions: 61 + (10 + 7 + 5) + 5 = 88. This matches, so 100 − 88 = 12 read none.

Answer: 61 people read exactly one newspaper.

Exam tips

Look for the keywords 'only', 'exactly', 'at least' and 'neither'. They decide which regions to add.
Always check whether the question gives the total population. If it does, there may be a 'none' region.
For three-set problems, start with the triple overlap. Everything else depends on it.
Verify by adding all regions. If the sum is not the total, find the error before choosing an option.
If a region comes out negative, the data is inconsistent or you forgot to subtract the triple. Recheck.

Practice questions from Set Theory, including Venn Diagram

Cardinality and Applications of Sets: frequently asked questions

What is the formula for n(A∪B∪C)?

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 pair overlaps, and add back the triple overlap.

How do I solve Venn diagram word problems quickly?

Draw the circles and fill the centre region first, then the pair-only regions, then the only-one regions. Finally find the 'none' region from the total. Check that all regions add up to the total.

What does n(A) mean in set theory?

n(A) is the cardinality of set A, which is the number of distinct elements in it. For example, if A = {1, 3, 5}, then n(A) = 3.

What is the difference between 'only A' and 'A'?

'A' includes every element of A, including those also in B or C. 'Only A' includes just the elements that are in A and in no other set.