Sets Word Problems

[MathDefinition title="Word Problems on Sets"] Word problems on sets involve translating real-life situations into set language and then solving using set operations like union, intersection, difference, and complement. [/MathDefinition]

[MathDefinition title="Cardinal Number of a Set"] The cardinal number of a set A, denoted by n(A), is the number of elements in set A. For finite sets, this is simply the count of elements. [/MathDefinition]

[MathKeyFormula title="Key Formulas for Word Problems"]

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(only A) = n(A) − n(A ∩ B)
4. n(neither A nor B) = n(U) − n(A ∪ B)
5. n(A' ∩ B') = n((A ∪ B)') [/MathKeyFormula]

[MathNote title="Solving Strategy"]

1. Identify the universal set and all given sets
2. Draw a Venn diagram if helpful
3. Fill in known values starting from intersections
4. Use formulas to find unknowns
5. Verify your answer makes sense in context [/MathNote]

[MathExample title="Example 1: Survey Problem"] Problem: In a survey of 500 people, 350 like tea, 250 like coffee, and 150 like both. How many people like neither tea nor coffee?

Solution: Let T = set of people who like tea, C = set of people who like coffee Given: n(U) = 500, n(T) = 350, n(C) = 250, n(T ∩ C) = 150

n(T ∪ C) = n(T) + n(C) − n(T ∩ C) = 350 + 250 − 150 = 450

People who like neither = n(U) − n(T ∪ C) = 500 − 450 = 50

Therefore, 50 people like neither tea nor coffee. [/MathExample]

[MathExample title="Example 2: Student Enrollment"] Problem: In a school, 120 students study Mathematics, 80 study Science, and 40 study both. If there are 200 students in total, how many students study neither Mathematics nor Science?

Solution: Let M = set of students studying Mathematics, S = set of students studying Science Given: n(U) = 200, n(M) = 120, n(S) = 80, n(M ∩ S) = 40

n(M ∪ S) = n(M) + n(S) − n(M ∩ S) = 120 + 80 − 40 = 160

Students studying neither = 200 − 160 = 40

Therefore, 40 students study neither subject. [/MathExample]

[MathExample title="Example 3: Newspaper Readership"] Problem: In a town of 1000 people, 600 read newspaper A, 400 read newspaper B, 200 read both. How many read newspaper A but not B?

Solution: Let A = set of people reading newspaper A, B = set of people reading newspaper B Given: n(A) = 600, n(B) = 400, n(A ∩ B) = 200

People reading A but not B = n(A − B) = n(A) − n(A ∩ B) = 600 − 200 = 400

Therefore, 400 people read newspaper A but not B. [/MathExample]

[MathExample title="Example 4: Language Speakers"] Problem: In a community of 800 people, 500 speak Hindi, 300 speak English, 200 speak both. Find the number of people who speak Hindi only.

Solution: Let H = set of Hindi speakers, E = set of English speakers Given: n(H) = 500, n(E) = 300, n(H ∩ E) = 200

People speaking Hindi only = n(H − E) = n(H) − n(H ∩ E) = 500 − 200 = 300

Therefore, 300 people speak Hindi only. [/MathExample]