How to Solve Two-Set Venn Diagram Problems?

Two-Set Venn Diagram Problems represent relationships between two groups using overlapping circles. This page explains union, intersection, elements in only one set, neither set and the universal set. It also gives formulas, calculation methods, shortcuts and solved numerical examples for quantitative aptitude questions.

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What Are Two-Set Venn Diagram Problems?

Two-set Venn diagram problems use two overlapping circles to show the relationship between sets A and B. The overlap represents elements common to both sets, written as A ∩ B.

The complete region covered by either circle represents A ∪ B. If the universal set contains N elements, the number outside both circles is N − n(A ∪ B). For example, if n(A) = 32, n(B) = 25 and n(A ∩ B) = 9, then n(A ∪ B) = 32 + 25 − 9 = 48.

Two-Set Venn Diagram Problems Video Lessons

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Venn Diagram Two-Set Problems

Learn to solve two-set Venn diagram survey problems using set relationships, intersections, unions, and the inclusion-exclusion principle.

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Two-Set Venn Diagram Problems Formula & Tricks

Important Formulas

Union of two sets
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

Add the sizes of A and B, then subtract the common elements because they were counted twice.

Neither A nor B
Neither = N − n(A ∪ B)

Subtract the union from the total number of elements in the universal set.

Exactly one set
Exactly one = n(A) + n(B) − 2n(A ∩ B)

This counts elements in A only and B only, excluding the overlap.

Only one set separately
A only = n(A) − n(A ∩ B); B only = n(B) − n(A ∩ B)

Remove the common region from each set to find its exclusive part.

At least one set
At least one = n(A ∪ B)

At least one means belonging to A, B or both.

Quick Tricks

Fill the overlap first

When separate-set totals and the common total are given, place the intersection first. Then subtract it from each set to obtain A only and B only.

Example: If n(A) = 40, n(B) = 28 and n(A ∩ B) = 12, then A only = 28 and B only = 16.
Use the complement for neither

For questions asking for neither, first calculate the union and subtract it from the universal-set total.

Example: If N = 100 and n(A ∪ B) = 73, then neither = 100 − 73 = 27.
Check for impossible values

The intersection cannot exceed the smaller set. Therefore, n(A ∩ B) ≤ min[n(A), n(B)]. Also, the union cannot exceed the universal-set total.

Example: If n(A) = 18 and n(B) = 25, an intersection of 21 is impossible because it is greater than n(A).

Two-Set Venn Diagram Problems Concepts

Intersection and Common Elements

The intersection A ∩ B contains elements that belong to both A and B.

If n(A ∩ B) = x, then x elements are placed in the overlapping part of the two circles. The intersection is included in both n(A) and n(B), so it must be subtracted once while finding the union.

Example: Among 50 students, 18 study Mathematics and 12 study both Mathematics and English. The 12 students in the intersection study both subjects.

Union and At Least One Set

The union A ∪ B contains every element belonging to A, B or both.

Use n(A ∪ B) = n(A) + n(B) − n(A ∩ B). The subtraction prevents the common elements from being counted twice.

Example: If 35 students study A, 27 study B and 10 study both, then at least one subject is studied by 35 + 27 − 10 = 52 students.

Only A, Only B and Exactly One

Only A excludes the overlap from A, while only B excludes the overlap from B.

The regions are A only = n(A) − n(A ∩ B) and B only = n(B) − n(A ∩ B). Exactly one set is the sum of these two regions.

Example: If n(A) = 30, n(B) = 24 and n(A ∩ B) = 8, then A only = 22, B only = 16 and exactly one = 38.

Neither Set and the Universal Set

Neither A nor B contains elements outside both circles but inside the universal set.

First calculate the union, then use neither = N − n(A ∪ B), where N is the total number of elements. If every element belongs to at least one set, the neither region is zero.

Example: In a group of 80 people, 46 like tea, 38 like coffee and 20 like both. The union is 64, so 16 people like neither beverage.

Two-Set Venn Diagram Problems Visual Guide

Use the diagram or chart below to understand the concept clearly.

Two-set Venn diagram with four regions

Visual guide: Two-set Venn diagram with four regions
The example has 18 elements in A only, 7 in both sets, 15 in B only and 10 in neither set.

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Practice Two-Set Venn Diagram Problems Questions

Practise published questions related to this topic.

1The price of petrol is increased by 28%. A person wants to increase his expenditure by 22% only. By approximately what percent should he decrease his consumption?→ 2An article is sold for Rs. 288 after successive discounts of 20% and 25%. What is the marked price of the article?→ 321.6 ÷ 3.6 × 2 + 0.25 × 16 ÷ 4 - 6 is equal to∶→ 4What is the value of x so that the seven digit number 55350x2 is divisible by 72?→ 5Two articles are sold for Rs. 4,752 each. On one, the seller gains 32% and on the other he loses 28%. What is his overall gain or loss percentage, correct to one decimal place?→ 6The price of sugar is increased by 17%. A person wants to increase his expenditure by 5% only. By approximately what percent should he decrease his consumption?→ 7The value of 15.2 + 5.8 ÷ 2.9 × 2 - 3.5 × 2 ÷ 0.5 is equal to:→ 8What is the value of x so that the seven digit number 8439x53 is divisible by 99?→ 9Two articles are sold for Rs. 962 each. On one, the seller gains 30% and on the other he loses 26%. What is his overall gain or loss percentage, nearest to one decimal place?→ 10An article is sold for Rs. 288 after successive discounts of 25% and x%. If the marked price of the article is Rs. 480, what is the value of x?→

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Quick Revision Notes

Two-Set Venn Diagram Revision Points

Use the set totals and overlap to calculate each region of a two-set Venn diagram.

  • n(A ∪ B) = n(A) + n(B) − n(A ∩ B).
  • A only = n(A) − n(A ∩ B).
  • B only = n(B) − n(A ∩ B).
  • Exactly one = n(A) + n(B) − 2n(A ∩ B).
  • Neither = N − n(A ∪ B).
  • At least one means the union; both means the intersection.
  • The intersection cannot be greater than either individual set.
  • The total of A only, intersection, B only and neither equals the universal-set total.

Two-Set Venn Diagram Problems FAQs

What is the formula for the union of two sets?

The formula is n(A ∪ B) = n(A) + n(B) − n(A ∩ B). The intersection is subtracted because common elements are counted in both set totals.

How do you find the number of people in neither set?

Use neither = N − n(A ∪ B). For example, if N = 90 and the union is 68, then 22 people belong to neither set.

How do you calculate exactly one of two sets?

Exactly one = n(A) + n(B) − 2n(A ∩ B). If the set sizes are 26 and 19 and the intersection is 6, exactly one is 26 + 19 − 12 = 33.

What is the difference between at least one and exactly one?

At least one includes the intersection, so it equals A ∪ B. Exactly one excludes the intersection and includes only A and only B.

If 45 people are in A, 32 are in B and 14 are in both, how many are in at least one set?

n(A ∪ B) = 45 + 32 − 14 = 63. Therefore, 63 people are in at least one set.

Can the intersection of two sets be greater than one of the sets?

No. The intersection must satisfy n(A ∩ B) ≤ n(A) and n(A ∩ B) ≤ n(B).

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