Venn Diagrams

[MathDefinition title="Venn Diagram"] A Venn diagram is a visual representation of sets using closed curves (usually circles) inside a rectangle (representing the universal set). The overlapping regions show common elements between sets. [/MathDefinition]

[MathDefinition title="Regions in a Two-Set Venn Diagram"] For two sets A and B within universal set U:

• Region I: Only A (A ∩ B')
• Region II: A ∩ B (common to both)
• Region III: Only B (A' ∩ B)
• Region IV: Neither A nor B (A' ∩ B') [/MathDefinition]

[MathKeyFormula title="Venn Diagram Formulas"]

1. n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
2. n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(C ∩ A) + n(A ∩ B ∩ C)
3. n(A') = n(U) − n(A)
4. n(only A) = n(A) − n(A ∩ B)
5. n(only B) = n(B) − n(A ∩ B) [/MathKeyFormula]

[MathNote title="Drawing Venn Diagrams"] Start with the innermost intersection and work outward. Always fill in the overlapping regions first, then the remaining parts of each set, and finally the region outside all sets but within U. [/MathNote]

[MathExample title="Example 1: Two-Set Venn Diagram"] Problem: In a class of 50 students, 30 like Mathematics and 25 like Science. If 15 like both subjects, find the number of students who like at least one subject.

Solution: Given: n(U) = 50, n(M) = 30, n(S) = 25, n(M ∩ S) = 15

Using the formula: n(M ∪ S) = n(M) + n(S) − n(M ∩ S) n(M ∪ S) = 30 + 25 − 15 n(M ∪ S) = 40

Therefore, 40 students like at least one subject. Students who like neither = 50 − 40 = 10 [/MathExample]

[MathExample title="Example 2: Three-Set Venn Diagram"] Problem: In a survey of 100 people, 60 read newspaper A, 40 read newspaper B, 30 read newspaper C, 20 read A and B, 15 read B and C, 10 read A and C, and 5 read all three. Find how many read none of the newspapers.

Solution: Given: n(A) = 60, n(B) = 40, n(C) = 30 n(A ∩ B) = 20, n(B ∩ C) = 15, n(A ∩ C) = 10 n(A ∩ B ∩ C) = 5

n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(C ∩ A) + n(A ∩ B ∩ C) = 60 + 40 + 30 − 20 − 15 − 10 + 5 = 100

Number who read none = 100 − 100 = 0 [/MathExample]

[MathExample title="Example 3: Finding Unknown Using Venn Diagram"] Problem: In a group of 200 people, 120 speak Hindi, 80 speak English, and 30 speak both. How many speak neither language?

Solution: Given: n(U) = 200, n(H) = 120, n(E) = 80, n(H ∩ E) = 30

n(H ∪ E) = n(H) + n(E) − n(H ∩ E) n(H ∪ E) = 120 + 80 − 30 = 170

Number who speak neither = n(U) − n(H ∪ E) = 200 − 170 = 30

Therefore, 30 people speak neither language. [/MathExample]

[MathExample title="Example 4: Venn Diagram with Three Sets"] Problem: In a survey of 60 students, 25 play cricket, 20 play football, 15 play hockey, 10 play cricket and football, 8 play football and hockey, 5 play cricket and hockey, and 3 play all three games. Find the number of students who play none of the games.

Solution: Given: n(C) = 25, n(F) = 20, n(H) = 15 n(C ∩ F) = 10, n(F ∩ H) = 8, n(C ∩ H) = 5 n(C ∩ F ∩ H) = 3

n(C ∪ F ∪ H) = 25 + 20 + 15 − 10 − 8 − 5 + 3 = 40

Students who play none = 60 − 40 = 20

Therefore, 20 students play none of the games. [/MathExample]