📚 Exercise 1.6 - Sets
Cardinality of Sets - Counting Elements using Formulas
1
If n(X) = 17, n(Y) = 23 and n(X ∪ Y) = 38, find n(X ∩ Y)
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1 Apply the Formula
Formula: n(X ∪ Y) = n(X) + n(Y) – n(X ∩ Y)
Substituting values:
38 = 17 + 23 – n(X ∩ Y)
38 = 40 – n(X ∩ Y)
Substituting values:
38 = 17 + 23 – n(X ∩ Y)
38 = 40 – n(X ∩ Y)
2 Solve for n(X ∩ Y)
n(X ∩ Y) = 40 – 38
n(X ∩ Y) = 2
n(X ∩ Y) = 2
The intersection has 2 elements
2
If n(X ∪ Y) = 18, n(X) = 8, n(Y) = 15, find n(X ∩ Y)
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1 Apply the Formula
Formula: n(X ∪ Y) = n(X) + n(Y) – n(X ∩ Y)
Substituting values:
18 = 8 + 15 – n(X ∩ Y)
18 = 23 – n(X ∩ Y)
Substituting values:
18 = 8 + 15 – n(X ∩ Y)
18 = 23 – n(X ∩ Y)
2 Solve for n(X ∩ Y)
n(X ∩ Y) = 23 – 18
n(X ∩ Y) = 5
n(X ∩ Y) = 5
The intersection has 5 elements
3
In a group of 400 people, 250 speak Hindi and 200 speak English. How many speak both?
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1 Define Sets
Let H = set of people who speak Hindi
Let E = set of people who speak English
n(H ∪ E) = 400 (total people)
n(H) = 250
n(E) = 200
Let E = set of people who speak English
n(H ∪ E) = 400 (total people)
n(H) = 250
n(E) = 200
2 Apply the Formula
n(H ∪ E) = n(H) + n(E) – n(H ∩ E)
400 = 250 + 200 – n(H ∩ E)
400 = 450 – n(H ∩ E)
400 = 250 + 200 – n(H ∩ E)
400 = 450 – n(H ∩ E)
3 Solve for n(H ∩ E)
n(H ∩ E) = 450 – 400
n(H ∩ E) = 50
n(H ∩ E) = 50
50 people speak both Hindi and English
Note: We assume each person speaks at least one of the two languages.
4
If n(S) = 21, n(T) = 32, n(S ∩ T) = 11, find n(S ∪ T)
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1 Apply the Formula
Formula: n(S ∪ T) = n(S) + n(T) – n(S ∩ T)
Substituting values:
n(S ∪ T) = 21 + 32 – 11
n(S ∪ T) = 53 – 11
Substituting values:
n(S ∪ T) = 21 + 32 – 11
n(S ∪ T) = 53 – 11
2 Calculate n(S ∪ T)
n(S ∪ T) = 42
The union S ∪ T has 42 elements
5
If n(X) = 40, n(X ∪ Y) = 60, n(X ∩ Y) = 10, find n(Y)
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1 Apply the Formula
Formula: n(X ∪ Y) = n(X) + n(Y) – n(X ∩ Y)
Substituting values:
60 = 40 + n(Y) – 10
60 = 30 + n(Y)
Substituting values:
60 = 40 + n(Y) – 10
60 = 30 + n(Y)
2 Solve for n(Y)
n(Y) = 60 – 30
n(Y) = 30
n(Y) = 30
Set Y has 30 elements
6
In a group of 70 people, 37 like coffee, 52 like tea. Each likes at least one drink. How many like both?
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1 Define Sets
Let C = set of people who like coffee
Let T = set of people who like tea
n(C ∪ T) = 70 (total people)
n(C) = 37
n(T) = 52
Let T = set of people who like tea
n(C ∪ T) = 70 (total people)
n(C) = 37
n(T) = 52
2 Apply the Formula
n(C ∪ T) = n(C) + n(T) – n(C ∩ T)
70 = 37 + 52 – n(C ∩ T)
70 = 89 – n(C ∩ T)
70 = 37 + 52 – n(C ∩ T)
70 = 89 – n(C ∩ T)
3 Solve for n(C ∩ T)
n(C ∩ T) = 89 – 70
n(C ∩ T) = 19
n(C ∩ T) = 19
19 people like both coffee and tea
7
In a group of 65 people, 40 like cricket, 10 like both cricket and tennis. Find tennis only and total tennis lovers
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1 Define Sets
Let C = set of people who like cricket
Let T = set of people who like tennis
n(C ∪ T) = 65 (total people)
n(C) = 40
n(C ∩ T) = 10
Let T = set of people who like tennis
n(C ∪ T) = 65 (total people)
n(C) = 40
n(C ∩ T) = 10
2 Find Total Tennis Lovers
n(C ∪ T) = n(C) + n(T) – n(C ∩ T)
65 = 40 + n(T) – 10
65 = 30 + n(T)
n(T) = 35
35 people like tennis
65 = 40 + n(T) – 10
65 = 30 + n(T)
n(T) = 35
35 people like tennis
3 Find Tennis Only (Not Cricket)
Tennis only = Total tennis – Both sports
n(T – C) = n(T) – n(C ∩ T)
n(T – C) = 35 – 10
n(T – C) = 25
25 people like tennis only and not cricket
n(T – C) = n(T) – n(C ∩ T)
n(T – C) = 35 – 10
n(T – C) = 25
25 people like tennis only and not cricket
• Tennis lovers: 35
• Tennis only (not cricket): 25
• Tennis only (not cricket): 25
8
In a committee, 50 speak French, 20 speak Spanish, 10 speak both. How many speak at least one?
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1 Define Sets
Let F = set of people who speak French
Let S = set of people who speak Spanish
n(F) = 50
n(S) = 20
n(F ∩ S) = 10
Let S = set of people who speak Spanish
n(F) = 50
n(S) = 20
n(F ∩ S) = 10
2 Find Union (At Least One Language)
Formula: n(F ∪ S) = n(F) + n(S) – n(F ∩ S)
Substituting values:
n(F ∪ S) = 50 + 20 – 10
n(F ∪ S) = 70 – 10
n(F ∪ S) = 60
Substituting values:
n(F ∪ S) = 50 + 20 – 10
n(F ∪ S) = 70 – 10
n(F ∪ S) = 60
60 people speak at least one of the two languages
💡 Key Insight: "At least one" means the union of the two sets. We add the individual counts and subtract those counted twice (the intersection).
