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Class 11 | CBSE | MATHEMATICS
SETS | EXERCISE 1.5

MATHEMATICS CLASS 11 :


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📘 Exercise 1.5 – Operations on Sets (Complement)

Class 11 | CBSE Mathematics

Question 1:

Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8} and C = {3, 4, 5, 6}. Find:
(i) A′
(ii) B′
(iii) (A ∪ C)′
(iv) (A ∪ B)′
(v) (A′)′
(vi) (B − C)′

(i) Finding A′

Step: Complement of A means all elements of U not in A.

A′ = U − A = {5, 6, 7, 8, 9}

✅ A′ = {5, 6, 7, 8, 9}

(ii) Finding B′

B′ = U − B = {1, 3, 5, 7, 9}

✅ B′ = {1, 3, 5, 7, 9}

(iii) Finding (A ∪ C)′

Step 1: Find the union

A ∪ C = {1, 2, 3, 4, 5, 6}

Step 2: Take complement

(A ∪ C)′ = {7, 8, 9}

✅ (A ∪ C)′ = {7, 8, 9}

(iv) Finding (A ∪ B)′

A ∪ B = {1, 2, 3, 4, 6, 8}

(A ∪ B)′ = {5, 7, 9}

✅ (A ∪ B)′ = {5, 7, 9}

(v) Finding (A′)′

Rule: Complement of complement returns the original set.

(A′)′ = A = {1, 2, 3, 4}

✅ (A′)′ = {1, 2, 3, 4}

(vi) Finding (B − C)′

Step 1: Find difference

B − C = {2, 8}

Step 2: Take complement

(B − C)′ = {1, 3, 4, 5, 6, 7, 9}

✅ (B − C)′ = {1, 3, 4, 5, 6, 7, 9}
🧠 Neurological Insight (Exam Memory Hack):
The brain processes Complement as “what is missing”. Always mentally fix the Universal Set first. This anchors working memory and prevents 90% of complement mistakes in exams.
🔑 1-Line Brain Rule:
A′ = U − A  |  (A′)′ = A

📘 Exercise 1.5 – Operations on Sets (Complement)

Class 11 | CBSE Mathematics

Question 2:

If U = {a, b, c, d, e, f, g, h}, find the complements of the following sets:

(i) A = {a, b, c}
(ii) B = {d, e, f, g}
(iii) C = {a, c, e, g}
(iv) D = {f, g, h, a}

(i) Finding A′

Mental Step: Remove elements of A from U.

A′ = U − A = {d, e, f, g, h}

✅ A′ = {d, e, f, g, h}

(ii) Finding B′

B′ = U − B = {a, b, c, h}

✅ B′ = {a, b, c, h}

(iii) Finding C′

Observation: C contains alternating letters.

C′ = U − C = {b, d, f, h}

✅ C′ = {b, d, f, h}

(iv) Finding D′

D′ = U − D = {b, c, d, e}

✅ D′ = {b, c, d, e}
🧠 Neurological Insight (High-Retention Rule):
The brain handles complements best when it performs a “visual deletion”. Always scan U once and mentally strike out the given set.
🔑 One-Line Exam Anchor:
Complement = What is LEFT in U after removal
⚠️ CBSE Examiner Tip:
Writing U − A before listing elements shows conceptual clarity and fetches full marks even if one element is missed.

📘 Exercise 1.5 – Complements of Sets

Class 11 | CBSE Mathematics

Question 3:

Taking the set of natural numbers ℕ = {1, 2, 3, …} as the universal set, write down the complements of the following sets:

(i) {x : x is an even natural number}
(ii) {x : x is an odd natural number}
(iii) {x : x is a positive multiple of 3}
(iv) {x : x is a prime number}
(v) {x : x is a natural number divisible by 3 and 5}
(vi) {x : x is a perfect square}
(vii) {x : x is a perfect cube}
(viii) {x : x + 5 = 8}
(ix) {x : 2x + 5 = 9}
(x) {x : x ≥ 7}
(xi) {x : x ∈ ℕ and 2x + 1 > 10}

(i) Complement of even natural numbers

Even numbers = {2, 4, 6, …}

✅ Complement = {x : x is an odd natural number}

(ii) Complement of odd natural numbers

✅ Complement = {x : x is an even natural number}

(iii) Complement of multiples of 3

Multiples of 3 = {3, 6, 9, …}

✅ Complement = {x : x ∈ ℕ and x is not divisible by 3}

(iv) Complement of prime numbers

✅ Complement = {1} ∪ {x : x is a composite natural number}

(v) Complement of numbers divisible by both 3 and 5

Divisible by 3 and 5 ⇒ divisible by 15

✅ Complement = {x : x ∈ ℕ and x is not divisible by 15}

(vi) Complement of perfect squares

Perfect squares = {1, 4, 9, 16, …}

✅ Complement = {x : x ∈ ℕ and x is not a perfect square}

(vii) Complement of perfect cubes

✅ Complement = {x : x ∈ ℕ and x is not a perfect cube}

(viii) Complement of {x : x + 5 = 8}

x + 5 = 8 ⇒ x = 3

✅ Complement = ℕ − {3}

(ix) Complement of {x : 2x + 5 = 9}

2x + 5 = 9 ⇒ x = 2

✅ Complement = ℕ − {2}

(x) Complement of {x : x ≥ 7}

✅ Complement = {1, 2, 3, 4, 5, 6}

(xi) Complement of {x : 2x + 1 > 10}

2x + 1 > 10 ⇒ x > 4.5 ⇒ x ≥ 5

✅ Complement = {1, 2, 3, 4}
🧠 Neurological Insight (Exam Survival Rule):
Always solve the condition first, then apply “Universal minus Answer”.
🔑 One-Line Brain Anchor:
Complement = Everything in ℕ that FAILS the rule
⚠️ CBSE Scoring Tip:
Writing answers in set-builder form is fully acceptable and avoids infinite listing errors.

📘 Exercise 1.5 – De Morgan’s Laws

Class 11 | CBSE Mathematics

Question 4:

If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}, verify that:

(i) (A ∪ B)′ = A′ ∩ B′
(ii) (A ∩ B)′ = A′ ∪ B′

(i) Verification of (A ∪ B)′ = A′ ∩ B′

Step 1: A ∪ B = {2, 3, 4, 5, 6, 7, 8}

Step 2: (A ∪ B)′ = U − (A ∪ B) = {1, 9}

Step 3: A′ = U − A = {1, 3, 5, 7, 9}

Step 4: B′ = U − B = {1, 4, 6, 8, 9}

Step 5: A′ ∩ B′ = {1, 9}

✅ (A ∪ B)′ = A′ ∩ B′ = {1, 9}
✔ Hence verified.

(ii) Verification of (A ∩ B)′ = A′ ∪ B′

Step 1: A ∩ B = {2}

Step 2: (A ∩ B)′ = U − {2} = {1, 3, 4, 5, 6, 7, 8, 9}

Step 3: A′ = {1, 3, 5, 7, 9}

Step 4: B′ = {1, 4, 6, 8, 9}

Step 5: A′ ∪ B′ = {1, 3, 4, 5, 6, 7, 8, 9}

✅ (A ∩ B)′ = A′ ∪ B′ = {1, 3, 4, 5, 6, 7, 8, 9}
✔ Hence verified.
🧠 Neurological Insight (Why De Morgan sticks):
The brain processes reversal patterns faster than direct logic.
Union flips to Intersection
Intersection flips to Union
under Complement.
🔑 One-Glance Exam Rule:
Complement → Change the symbol + Complement each set
⚠️ CBSE Scoring Hack:
Writing both LHS and RHS separately and matching them guarantees full marks even if arithmetic slips.

📘 Exercise 1.5 – Venn Diagram Representation

Class 11 | CBSE Mathematics

Question 5:

Draw appropriate Venn diagrams for each of the following:
(i) (A ∪ B)′
(ii) A′ ∩ B′
(iii) (A ∩ B)′
(iv) A′ ∪ B′

(i) Venn diagram for (A ∪ B)′

A ∪ B includes all elements lying in set A, set B, and their overlapping region.

✅ Shade the region of the universal set outside both A and B.
🧠 Brain Cue:
Union fills everything → Complement removes everything inside.
Only the outside area survives.

(ii) Venn diagram for A′ ∩ B′

A′ means outside A and B′ means outside B.

✅ Shade only the region which lies outside both A and B.
🧠 Neural Match:
Outside A ∩ Outside B → same shaded area as (A ∪ B)′

(iii) Venn diagram for (A ∩ B)′

A ∩ B is the common overlapping region of A and B.

✅ Shade all regions except the overlapping part.
🧠 Visual Shock:
Brain deletes only the middle overlap — everything else stays shaded.

(iv) Venn diagram for A′ ∪ B′

A′ = outside A, B′ = outside B.

✅ Shade all regions except the common intersection of A and B.
🧠 Pattern Lock:
Union of complements = everything except overlap.
Same picture as (A ∩ B)′.
🧠 De Morgan’s Visual Truth (Exam Gold):
(A ∪ B)′ and A′ ∩ B′ always have identical shading.

(A ∩ B)′ and A′ ∪ B′ always have identical shading.
🔑 30-Second Diagram Rule:
Complement = Erase the named region and shade everything else.
⚠️ CBSE Scoring Tip:
Even rough hand-drawn Venn diagrams with correct shading fetch full marks.

📘 Exercise 1.5 – Complement of a Set

Class 11 | CBSE Mathematics

Question 6:

Let U be the set of all triangles in a plane. If A is the set of all triangles with at least one angle different from 60°, what is A′?

Understanding the Given Condition

Every triangle has three interior angles whose sum is 180°.

If a triangle has at least one angle different from 60°, then it is not an equilateral triangle.

✅ Set A consists of all non-equilateral triangles.

Finding the Complement A′

The complement A′ contains all triangles not belonging to A.

That means triangles in which no angle is different from 60°.

Hence, all three angles must be exactly 60°.

✅ A′ is the set of all equilateral triangles.
🧠 Neurological Insight (Concept Lock):
The brain processes negative conditions by reversal.
“At least one angle ≠ 60°” flips to
“All angles = 60°”.
🔑 One-Line Brain Anchor:
Complement = Exact opposite condition
⚠️ CBSE Examiner Tip:
Writing the answer in words is fully correct here. Mathematical symbols are not compulsory.

📘 Exercise 1.5 – Laws of Complement

Class 11 | CBSE Mathematics

Question 7:

Fill in the blanks to make each of the following a true statement:

(i) A ∪ A′ = ……
(ii) φ′ ∩ A = ……
(iii) A ∩ A′ = ……
(iv) U′ ∩ A = ……

(i) A ∪ A′

A set together with its complement covers every element of the universal set.

✅ A ∪ A′ = U

(ii) φ′ ∩ A

The complement of the empty set is the universal set.

φ′ = U

✅ φ′ ∩ A = A

(iii) A ∩ A′

A set and its complement have no common elements.

✅ A ∩ A′ = φ

(iv) U′ ∩ A

The complement of the universal set is the empty set.

U′ = φ

✅ U′ ∩ A = φ
🧠 Neurological Insight (4 Laws – 4 Brain Locks):
Union with complement → Everything
Intersection with complement → Nothing
🔑 One-Line Memory Code:
A + NOT A = U
A × NOT A = φ
⚠️ CBSE Rapid-Score Tip:
These four results are standard identities. Writing them directly saves time and ensures full marks.

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