Universal Set – Class 11 Applied Mathematics

Definition of Universal Set

A Universal Set is a set that contains all the elements under consideration for a particular context or problem. It is typically denoted by the symbol U. Every other set in the given context is a subset of the universal set.

U = {x | x ∈ S for all sets S in the context}

Example: Solved Example: Identifying U

Show Step-by-Step Solution

Step 1: Consider sets A = {1, 2, 3} and B = {3, 4, 5}.
Step 2: Identify all unique elements present in both sets.
Step 3: Define U = {1, 2, 3, 4, 5}.
Answer: U = {1, 2, 3, 4, 5}

Key Properties of Universal Sets

The universal set acts as the parent set for all subsets in a specific discussion. For any set A, the complement of A, denoted as A', is defined as U - A. The union of any set A with its complement A' always results in the universal set U.

A ∪ A' = U

Example: Solved Example: Verifying A ∪ A'

Show Step-by-Step Solution

Step 1: Let U = {1, 2, 3, 4, 5, 6} and A = {1, 2, 3}.
Step 2: Find A' = U - A = {4, 5, 6}.
Step 3: Calculate A ∪ A' = {1, 2, 3} ∪ {4, 5, 6} = {1, 2, 3, 4, 5, 6}.
Answer: A ∪ A' = U

Universal Set in Venn Diagrams

In a Venn diagram, the universal set is represented by a rectangle. All other subsets are represented by circles drawn inside this rectangle. The region inside the rectangle but outside the circles represents the elements of U that are not in the subsets.

n(U) = n(A) + n(A')

Example: Solved Example: Calculating n(A')

Show Step-by-Step Solution

Step 1: Given n(U) = 50 and n(A) = 20.
Step 2: Use the formula n(A') = n(U) - n(A).
Step 3: n(A') = 50 - 20 = 30.
Answer: n(A') = 30

Section A: Multiple Choice Questions (MCQs)

easyIf U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {2, 4, 6, 8, 10}, find A'.

A. {1, 3, 5, 7, 9} B. {2, 4, 6, 8, 10} C. {1, 2, 3, 4, 5} D. {6, 7, 8, 9, 10}

Show Answer

✔ Correct: A

A' = U - A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} - {2, 4, 6, 8, 10} = {1, 3, 5, 7, 9}.

easyIf U = {x : x is a natural number ≤ 10} and A = {x : x is a prime number ≤ 10}, find n(A').

A. 4 B. 6 C. 5 D. 7

Show Answer

✔ Correct: B

U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {2, 3, 5, 7}. A' = {1, 4, 6, 8, 9, 10}. n(A') = 6.

mediumGiven U = {a, b, c, d, e, f, g, h}, A = {a, b, c}, B = {d, e, f}. Find (A ∪ B)'.

A. {g, h} B. {a, b, c, d, e, f} C. {a, b, c, g, h} D. {d, e, f, g, h}

Show Answer

✔ Correct: A

A ∪ B = {a, b, c, d, e, f}. (A ∪ B)' = U - {a, b, c, d, e, f} = {g, h}.

mediumIf n(U) = 100, n(A) = 40, n(B) = 30, and n(A ∩ B) = 10, find n((A ∪ B)').

A. 30 B. 40 C. 50 D. 60

Show Answer

✔ Correct: B

n(A ∪ B) = n(A) + n(B) - n(A ∩ B) = 40 + 30 - 10 = 60. n((A ∪ B)') = 100 - 60 = 40.

hardLet U = {1, 2, ..., 20}. If A = {x : x is a multiple of 3} and B = {x : x is a multiple of 4}, find n(A ∩ B)'.

A. 18 B. 19 C. 17 D. 20

Show Answer

✔ Correct: A

A = {3, 6, 9, 12, 15, 18}, B = {4, 8, 12, 16, 20}. A ∩ B = {12}. n(A ∩ B) = 1. n(A ∩ B)' = 20 - 1 = 19. Wait, checking: A ∩ B = {12}. n(A ∩ B)' = 20 - 1 = 19.

hardIf U = {x : x ∈ ℤ, 1 ≤ x ≤ 10}, A = {x : x is even}, B = {x : x is a multiple of 3}. Find n(A' ∩ B').

A. 3 B. 4 C. 5 D. 2

Show Answer

✔ Correct: A

A' ∩ B' = (A ∪ B)'. A = {2, 4, 6, 8, 10}, B = {3, 6, 9}. A ∪ B = {2, 3, 4, 6, 8, 9, 10}. (A ∪ B)' = {1, 5, 7}. Count = 3.

very hardLet U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. If A = {x : x² - 5x + 6 = 0} and B = {x : x² - 7x + 12 = 0}, find n((A ∪ B)').

A. 6 B. 7 C. 8 D. 9

Show Answer

✔ Correct: C

A = {2, 3}, B = {3, 4}. A ∪ B = {2, 3, 4}. (A ∪ B)' = {1, 5, 6, 7, 8, 9, 10}. Count = 8.

Section B: Assertion–Reason Questions

Assertion (A): If U = {1, 2, 3} and A = {1, 2}, then A' = {3}.
Reason (R): The complement of a set A with respect to U is the set of elements in U that are not in A.

Choose the correct option:
(A) Both A and R are true and R is the correct explanation of A.
(B) Both A and R are true but R is not the correct explanation of A.
(C) A is true but R is false.
(D) A is false but R is true.

A. Option A B. Option B C. Option C D. Option D

Show Answer

✔ Correct: A

Both assertion and reason are true, and the reason correctly explains the assertion.

Assertion (A): For any set A ⊆ U, A ∩ A' = U.
Reason (R): The intersection of a set and its complement is always the empty set.

Choose the correct option:
(A) Both A and R are true and R is the correct explanation of A.
(B) Both A and R are true but R is not the correct explanation of A.
(C) A is true but R is false.
(D) A is false but R is true.

A. Option A B. Option B C. Option C D. Option D

Show Answer

✔ Correct: D

Assertion is false because A ∩ A' = ∅, not U. Reason is true.

Section C: Short Answer Questions

mediumGiven U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8}. Find: (i) A' (ii) B' (iii) (A ∪ B)'.

Show Answer & Solution

(i) A' = {5, 6, 7, 8, 9}
(ii) B' = {1, 3, 5, 7, 9}
(iii) A ∪ B = {1, 2, 3, 4, 6, 8}. (A ∪ B)' = {5, 7, 9}

mediumGiven U = {x : x is a digit}, A = {1, 3, 5, 7}, B = {0, 2, 4, 6}. Find: (i) n(A) (ii) n(B') (iii) n(A ∩ B)'.

Show Answer & Solution

(i) n(A) = 4
(ii) B' = {1, 3, 5, 7, 8, 9}, n(B') = 6
(iii) A ∩ B = ∅, (A ∩ B)' = U, n(U) = 10

hardLet U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. If A = {x : x is prime}, B = {x : x is even}. Find: (i) A ∩ B (ii) (A ∩ B)' (iii) A' ∪ B'.

Show Answer & Solution

(i) A = {2, 3, 5, 7}, B = {2, 4, 6, 8, 10}. A ∩ B = {2}
(ii) (A ∩ B)' = {1, 3, 4, 5, 6, 7, 8, 9, 10}
(iii) A' ∪ B' = (A ∩ B)' = {1, 3, 4, 5, 6, 7, 8, 9, 10}

Section D: Long Answer Questions

hardIn a survey of 50 people, 30 like tea, 25 like coffee, and 10 like both. Find: (i) How many like only tea? (ii) How many like only coffee? (iii) How many like neither? (iv) Verify n(U) = n(T ∪ C) + n((T ∪ C)'). (v) Calculate n((T ∩ C)').

Show Answer & Solution

Step 1: Given n(U)=50, n(T)=30, n(C)=25, n(T ∩ C)=10.
Step 2: Only tea = n(T) - n(T ∩ C) = 30 - 10 = 20.
Step 3: Only coffee = n(C) - n(T ∩ C) = 25 - 10 = 15.
Step 4: n(T ∪ C) = 20 + 15 + 10 = 45. Neither = 50 - 45 = 5.
Step 5: n((T ∩ C)') = 50 - 10 = 40.

very hardLet U = {1, 2, ..., 12}. Let A = {x : x is a multiple of 2}, B = {x : x is a multiple of 3}, C = {x : x is a multiple of 4}. Find: (i) A ∩ B ∩ C (ii) (A ∪ B ∪ C)' (iii) n(A ∪ B ∪ C) (iv) Verify De Morgan's Law (A ∪ B)' = A' ∩ B' (v) Calculate n(A' ∩ B' ∩ C').

Show Answer & Solution

Step 1: A={2,4,6,8,10,12}, B={3,6,9,12}, C={4,8,12}.
Step 2: A ∩ B ∩ C = {12}.
Step 3: A ∪ B ∪ C = {2,3,4,6,8,9,10,12}. (A ∪ B ∪ C)' = {1,5,7,11}.
Step 4: n(A ∪ B ∪ C) = 8.
Step 5: A' ∩ B' ∩ C' = (A ∪ B ∪ C)' = {1,5,7,11}, count = 4.

Section E: Case-Based Question

★

Case Study: A school library tracks 100 students. 60 students borrowed a Science book, 40 borrowed a Math book, and 20 borrowed both. Let U be the set of all students, S be the set of Science borrowers, and M be the set of Math borrowers.

Sub-question 1: (i) Find the number of students who borrowed only Science books. (ii) Find the number of students who borrowed neither.

Show Solution

(i) Only Science = n(S) - n(S ∩ M) = 60 - 20 = 40. (ii) n(S ∪ M) = 60 + 40 - 20 = 80. Neither = 100 - 80 = 20.

Sub-question 2: If 10 students who borrowed neither are now added to the Math group, find the new n(M') and n(S ∩ M).

Show Solution

New n(M) = 40 + 10 = 50. New n(M') = 100 - 50 = 50. New n(S ∩ M) = 20 + 10 = 30.

CBSE Board & Sample Paper Past Year Questions

📋 CBSE 2023 📝 1 Mark

If U = {1, 2, 3, 4, 5} and A = {1, 2}, find A'.

Show Answer & Solution

A' = U - A = {3, 4, 5}.

📋 CBSE 2022 📝 1 Mark

Given n(U) = 50, n(A) = 20, find n(A').

Show Answer & Solution

n(A') = n(U) - n(A) = 50 - 20 = 30.

📋 CBSE Sample Paper 2024 📝 1 Mark

If A ⊂ U, then A ∪ A' is equal to:

Show Answer & Solution

✔ Answer: A

By definition, the union of a set and its complement is the universal set U.

📋 CBSE 2020 📝 2 Marks

If U = {x : x is a natural number ≤ 6} and A = {2, 4, 6}, find A'.

Show Answer & Solution

U = {1, 2, 3, 4, 5, 6}. A = {2, 4, 6}. A' = {1, 3, 5}.

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