Venn Diagrams Concepts

1. Two-Set Venn Diagram

• Formula:

$$n(A\cup B)=n(A)+n(B)-n(A\cap B)$$ $$n(\text{only }A)=n(A)-n(A\cap B)$$ $$n(\text{only }B)=n(B)-n(A\cap B)$$ $$n(\text{neither})=N-n(A\cup B)$$

• Example: In a group of 200 students, 120 know Python and 100 know Java. If 60 know both, find how many know only Python and neither. Solution: Only Python:

$$120-60=\boxed{60}$$

Total using at least one:

$$120+100-60=160$$

Neither:

$$200-160=\boxed{40}$$

---

2. Three-Set Venn Diagram

• Formula:

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

• Example: In a group of 200 people, 100 use Python, 80 use Java, 70 use C++. The pairwise intersections are 40, 30 and 25, while 10 use all three. Find the number using at least one language. Solution:

$$n(A\cup B\cup C) =100+80+70-40-30-25+10$$ $$=235-95+10 =\boxed{150}$$

---

3. Only One Category in Three Sets

• Formula:

$$n(\text{only }A) =n(A)-n(A\cap B)-n(A\cap C)+n(A\cap B\cap C)$$

Similarly for only B and only C.

• Example: 150 people are surveyed. 70 like cricket, 60 football and 50 tennis. (30) like cricket and football, (25) like cricket and tennis, and (10) like all three. Find those who like only cricket. Solution:

$$\text{Only cricket} =70-30-25+10$$ $$=\boxed{25}$$

---

4. Exactly Two Categories

• Formula:

$$n(\text{exactly two}) =n(A\cap B)+n(B\cap C)+n(C\cap A) -3n(A\cap B\cap C)$$

If pairwise intersections include the three-way intersection, subtract the triple intersection once from each pairwise-only region.

• Example: 40 like A and B, 30 like B and C, 25 like C and A, and 10 like all three. Find the number who like exactly two categories. Solution:

$$40+30+25-3(10)$$ $$=95-30 =\boxed{65}$$

---

5. Exactly One Category

• Formula:

$$n(\text{exactly one}) =n(A)+n(B)+n(C) -2[n(A\cap B)+n(B\cap C)+n(C\cap A)] +3n(A\cap B\cap C)$$

• Example: 80 like A, 70 like B and 60 like C. Pairwise intersections are 30, 25 and 20, and 10 like all three. Find the number who like exactly one. Solution:

$$80+70+60-2(30+25+20)+3(10)$$ $$=210-150+30 =\boxed{90}$$

---

6. None of the Categories

• Formula:

$$n(\text{none})=N-n(A\cup B\cup C)$$

• Example: In a group of 300 people, 180 like A, 150 like B, 120 like C. Pairwise intersections are 70, 50 and 60, and 30 like all three. Find how many like none. Solution:

$$n(A\cup B\cup C) =180+150+120-70-50-60+30$$ $$=300$$

Therefore:

$$n(\text{none})=300-300=\boxed{0}$$

---

7. Two-Set Maximum and Minimum Overlap

• Formula:

$$\text{Maximum overlap}=\min(n(A),n(B))$$ $$\text{Minimum overlap} =\max(0,n(A)+n(B)-N)$$

• Example: In a class of 180 students, 100 passed Maths and 90 passed Science. Find the maximum and minimum possible number who passed both. Solution: Maximum:

$$\min(100,90)=\boxed{90}$$

Minimum:

$$\max(0,100+90-180) =\boxed{10}$$

---

8. Three-Set Maximum/Minimum Overlap

• Formula: For all three sets:

$$\max(A\cap B\cap C)=\min(A,B,C)$$

The minimum possible triple overlap is:

$$\max(0,A+B+C-2N)$$

when only the set totals and overall population are known.

• Example: In a group of 100, 60 like A, 50 like B and 40 like C. Find the maximum and minimum possible number who like all three. Solution: Maximum:

$$\min(60,50,40)=\boxed{40}$$

Minimum:

$$\max(0,60+50+40-200) =\max(0,-50) =\boxed{0}$$

---

9. Missing Value Reconstruction

• Formula: Use:

$$\text{Total}=\text{Only regions}+\text{pairwise regions}+\text{triple intersection}+\text{none}$$

Hence:

$$\text{Missing region} =\text{Total}-\text{sum of all known regions}$$

• Example: In a survey of 100 people, 30 use only A, 20 only B, 15 only C, 10 use A and B only, 5 use B and C only, 8 use C and A only, and 4 use all three. Find the number using none. Solution: People using at least one:

$$30+20+15+10+5+8+4=92$$

Therefore:

$$\text{None}=100-92 =\boxed{8}$$

---

10. Converting Total Set Counts to Venn Regions

• Formula: If (n(A)) includes all regions inside A:

$$\text{Only }A=n(A)-(\text{A∩B only})-(\text{A∩C only})-(A\cap B\cap C)$$

If pairwise intersection includes the triple intersection:

$$\text{Only }A=n(A)-n(A\cap B)-n(A\cap C)+n(A\cap B\cap C)$$

• Example: 100 students study A, 70 study B, 50 study C. If 30 study A and B, 20 study A and C, and 10 study all three, find only A. Solution:

$$\text{Only A}=100-30-20+10$$ $$=\boxed{60}$$

---

11. Survey Data with Percentages

• Formula:

$$\text{Number}=\frac{\text{Percentage}}{100}\times\text{Total}$$

Then apply the appropriate Venn diagram formula.

• Example: In a survey of 500 people, 60% use A, 40% use B, and 20% use both. Find the percentage using only A. Solution: Only A:

$$60$$

Number:

$$500\times\frac{40}{100} =\boxed{200}$$

---

12. At Least One Category

• Formula:

$$n(\text{at least one})=n(A\cup B\cup C)$$

Therefore:

$$n(\text{none})=N-n(A\cup B\cup C)$$

• Example: In a group of 250 students, 150 study Java, 100 study Python, and 50 study both. How many study at least one? Solution:

$$n(J\cup P)=150+100-50$$ $$=\boxed{200}$$

---

13. Set Difference / Only One of Two

• Formula:

$$n(A-B)=n(A)-n(A\cap B)$$ $$n(B-A)=n(B)-n(A\cap B)$$

• Example: 80 employees know SQL, 50 know both SQL and Python. Find how many know SQL but not Python. Solution:

$$80-50=\boxed{30}$$

---

14. Data Interpretation from Tables

• Formula: For row/column percentage questions:

$$\text{Percentage}=\frac{\text{Required value}}{\text{Base value}}\times100$$

For percentage increase:

$$\frac{\text{New}-\text{Old}}{\text{Old}}\times100$$

• Example: A company has 240 employees in 2025 and 300 in 2026. Find the percentage increase. Solution: Increase:

$$300-240=60$$

Therefore:

\frac{60}{240}\times100 =\boxed{25%}

---

15. Ratio-Based Data Interpretation

• Formula: If quantities are in ratio (a:b) and total is (T):

$$A=\frac{a}{a+b}T,\qquad B=\frac{b}{a+b}T$$

• Example: The numbers of male and female employees are in the ratio (3:2). If there are 500 employees, find the number of females. Solution: Total parts:

$$3+2=5$$

Female employees:

$$\frac25\times500 =\boxed{200}$$

---

Advanced Variants

16. Four-Set Inclusion-Exclusion

• Formula:

$$n(A\cup B\cup C\cup D) =\sum n(A_i)-\sum n(A_i\cap A_j) +\sum n(A_i\cap A_j\cap A_k) -n(A\cap B\cap C\cap D)$$

• Example: Four groups contain 50, 40, 30 and 20 members. The sum of all pairwise intersections is 35, the sum of all triple intersections is 8, and all four overlap in 2. Find the number in at least one group. Solution:

$$50+40+30+20-35+8-2$$ $$=140-35+8-2 =\boxed{111}$$

---

17. Maximum Number in Exactly Two Sets

• Formula: To maximize a specific overlap (A\cap B), use:

$$\max(A\cap B)=\min(n(A),n(B))$$

subject to all stated set and population constraints.

• Example: In a group of 100, 70 like A and 60 like B. What is the maximum possible number who like both? Solution: The overlap cannot exceed the smaller set:

$$\min(70,60) =\boxed{60}$$

---

18. Minimum Number in At Least One of Three Sets

• Formula: To minimize the union, maximize the overlaps while respecting set sizes. For three sets:

$$n(A\cup B\cup C)\ge \max(n(A),n(B),n(C))$$

• Example: In a class, 60 students like A, 50 like B and 40 like C. What is the minimum possible number who like at least one? Solution: All smaller groups can be contained within the largest group.

Therefore minimum union:

$$\boxed{60}$$

---

19. Maximum Number in At Least One of Three Sets

• Formula:

$$n(A\cup B\cup C)\le\min(N,n(A)+n(B)+n(C))$$

The upper bound occurs when overlaps are minimized.

• Example: In a group of 200, 70 like A, 60 like B and 50 like C. Find the maximum possible number who like at least one. Solution:

$$70+60+50=180$$

Since (180<200), the sets can be completely disjoint.

Therefore:

$$\boxed{180}$$

---

20. Consistency Check in Venn Diagram Data

• Formula: Every Venn region must be non-negative:

$$\text{Only A}=A-AB-AC+ABC\ge0$$

Similar conditions apply to every region.

• Example: (n(A)=50,\ n(A\cap B)=30,\ n(A\cap C)=25,\ n(A\cap B\cap C)=5). Is the data valid? Solution:

$$\text{Only A}=50-30-25+5=0$$

This is non-negative, so the A-related data is consistent.

$$\boxed{\text{Data is possible}}$$