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13. Sets

CSS GSA — Paper 7, Part-II, Section VI, Sub-topic 13 (Sets)

Past Paper Weightage

  • Moderate — Venn diagram/union-intersection word problems appear regularly (marked as a focus area for revision).

Key Concepts

  • Set: a well-defined collection of distinct objects (elements)
  • Union (A∪B): all elements in A or B or both; Intersection (A∩B): elements common to both A and B
  • Complement (A′): elements in the universal set but not in A
  • Difference (A−B): elements in A but not in B
  • Subset (A⊆B): every element of A is also in B
  • Empty set (∅): contains no elements; Universal set (U): contains all elements under consideration
  • Venn diagrams: visual tool for set operations and overlapping-group word problems

Formula (2-set problems)

n(A∪B)=n(A)+n(B)−n(A∩B)n(A\cup B) = n(A) + n(B) - n(A\cap B)

essential for "how many people like both/either/neither" word problems

For 3 sets:

n(A∪B∪C)=n(A)+n(B)+n(C)−n(A∩B)−n(B∩C)−n(A∩C)+n(A∩B∩C)n(A\cup B\cup C) = n(A)+n(B)+n(C) - n(A\cap B) - n(B\cap C) - n(A\cap C) + n(A\cap B\cap C)

Worked Example

  • In a class of 50, 30 like Math, 25 like Science, 15 like both. How many like at least one?

    n(A∪B) = 30+25−15 = 40

Past Paper Style Numericals (Solved)

  1. In a group of 70 students, 45 play cricket, 30 play football, and 12 play both. How many play neither?

    Answer: n(C∪F)=45+30−12=63; neither = 70−63 = 7

  2. Out of 120 people, 65 like tea only, 40 like coffee only, and the rest like both. How many like both?

    Answer: both = 120−(65+40) = 15

  3. If U={1..10}, A={2,4,6,8,10}, B={1,2,3,4,5}, find A∩B and A∪B.

    Answer: A∩B={2,4}; A∪B={1,2,3,4,5,6,8,10}

  4. In a class of 50, every student takes Math or Physics or both; 32 take Math, 25 take Physics. How many take both?

    Answer: 32+25−50 = 7

MCQ Traps

  • "At least one" = union; "both" = intersection; "neither" = complement of the union — precise wording changes the required set operation
  • Don't forget to subtract the intersection once when using the union formula (double-counting is the most common error)

Self-Test Questions

  1. If n(A)=40, n(B)=35, n(A∩B)=15, find n(A∪B).
  2. In a survey of 100 people, 60 read newspaper X, 50 read newspaper Y, 20 read both. How many read neither?
  3. Define subset and give an example using two small sets.

Past-Paper MCQs (FPSC)

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The FPSC MCQ papers you shared (2024 and 2025) had no direct sets question.

Past-Paper Questions with Answers (FPSC)

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Every question below appeared in the FPSC papers you shared; the year and question number are in each title. Answers are worked out by me.

  • CSS 2019 Q8(b): U = whole numbers from 10 to 24, A = even numbers, B = numbers divisible by 5. Write the number of elements of A∩B. (5)
    1. U = {10, 11, …, 24}
    2. A = {10, 12, 14, 16, 18, 20, 22, 24}
    3. B = {10, 15, 20}
    4. A∩B = {10, 20}, so n(A∩B) = 2
  • CSS 2019 Q8(c): In the Venn diagram, A = Americans, S = scientists, P = politicians. Find (i) Americans who are politicians but not scientists, (ii) scientists who are politicians but not Americans. (5)
    • Regions: a = A∩S∩P; b = A∩P only; c = A only; d = A∩S only; e = S only; f = S∩P only; g = P only
    • (i) American politicians but not scientists = region b
    • (ii) Scientist politicians but not Americans = region f

CSS Exam Perspective

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  • Most repeated areas: intersection of conditions (2019), reading a three-circle Venn diagram
  • Tip: list the sets in full before intersecting; shade the required region lightly