Venn Diagrams and Inclusion-Exclusion Principle | NIMCET

Quick answer: For two sets, $n(A \cup B) = n(A) + n(B) - n(A \cap B)$. For three sets, $n(A \cup B \cup C) = \sum n(A) - \sum n(A \cap B) + n(A \cap B \cap C)$. This is the inclusion-exclusion principle: add the single sets, subtract the overlaps counted twice, then add back the triple overlap. A Venn diagram filled from the centre outward makes these problems quick and reliable.

This uses the operations in set operations and laws of sets.

Two-set formulas

If A and B are disjoint, $n(A \cup B) = n(A) + n(B)$.

Three-set formula

$n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(C \cap A) + n(A \cap B \cap C)$

Let $S_1 = n(A) + n(B) + n(C)$, $S_2$ = sum of the three pairwise intersections, and $S_3 = n(A \cap B \cap C)$.

How to fill a three-set Venn diagram

Step 1: Write $n(A \cap B \cap C)$ in the centre.

Step 2: For each pair, write (pair intersection − centre) in the region of exactly those two sets.

Step 3: For each set, write (set total − its three inner regions) in the "only" region.

Step 4: Add all seven regions for the union, and subtract from $n(U)$ for "none".

Worked examples

Example 1: two languages

In a class of 50, 30 study French, 25 study German and 10 study both. How many study neither?

Step 1: $n(F \cup G) = 30 + 25 - 10 = 45$.

Step 2: Neither $= 50 - 45 = 5$.

Example 2: three newspapers

In a survey of 100 people: 50 read A, 40 read B, 30 read C, 20 read A and B, 15 read B and C, 10 read A and C, and 5 read all three.

At least one:

$120 - 45 + 5 = 80$

None: $100 - 80 = 20$.

Exactly one: $S_1 - 2S_2 + 3S_3 = 120 - 90 + 15 = 45$.

Exactly two: $S_2 - 3S_3 = 45 - 15 = 30$.

Check: exactly one + exactly two + exactly three $= 45 + 30 + 5 = 80$ ✓.

Example 3: region-by-region

Using Example 2, the "only A" region is $50 - (20 - 5) - (10 - 5) - 5 = 25$.

Example 4: the largest and smallest possible overlap

If $n(U) = 100$, $n(A) = 70$ and $n(B) = 60$, what are the possible values of $n(A \cap B)$?

• Maximum: $\min(70, 60) = 60$, when B ⊆ A.
• Minimum: $70 + 60 - 100 = 30$, when $A \cup B = U$.

So $30 \le n(A \cap B) \le 60$.

Summary table

Exam tips

• Always fill a Venn diagram from the centre outward.
• Read wording carefully: "both A and B" may include those who also do C.
• Check that the seven regions plus "none" add up to the total.
• The same principle is used for counting with divisibility, for example numbers divisible by 2, 3 or 5.

The same counting idea appears in probability; see NIMCET probability and statistics and the NIMCET Set Theory chapter.

Common mistakes

• Subtracting the triple intersection instead of adding it back.
• Treating a pairwise intersection as "exactly two"; it includes the centre.
• Forgetting the "none" region outside all circles.
• Using $n(A) + n(B)$ for a union without checking whether the sets overlap.

Frequently Asked Questions

What is the inclusion-exclusion principle?

It is the counting rule n(A ∪ B) = n(A) + n(B) − n(A ∩ B), and its extension to more sets, which corrects for elements counted more than once.

What is the formula for n(A ∪ B ∪ C)?

n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(C ∩ A) + n(A ∩ B ∩ C).

How do you find the number in exactly one of three sets?

Use S₁ − 2S₂ + 3S₃, where S₁ is the sum of the set sizes, S₂ the sum of pairwise intersections and S₃ the triple intersection.

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

Subtract the union from the total: n(U) − n(A ∪ B).

What is the minimum possible value of n(A ∩ B)?

It is n(A) + n(B) − n(U), or 0 if that value is negative.