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Class 11 | CBSE | MATHEMATICS
SETS | EXERCISE 1.4

MATHEMATICS CLASS 11 :

Union of Sets – Solved Problems

Golden Rule (Burn this into memory):
The union of two sets contains all elements belonging to either set or both, but no repetition.

A ∪ B = { x : x ∈ A or x ∈ B }

1. Find the union of each of the following pairs of sets :
(i) X = {1, 3, 5} Y = {1, 2, 3}
(ii) A = [ a, e, i, o, u} B = {a, b, c}
(iii) A = {x : x is a natural number and multiple of 3}
B = {x : x is a natural number less than 6}
(iv) A = {x : x is a natural number and 1 < x ≤6 }
B = {x : x is a natural number and 6 < x < 10 }
(v) A = {1, 2, 3}, B = φ

(i) X ∪ Y

X = {1, 3, 5}
Y = {1, 2, 3}

X ∪ Y = {1, 2, 3, 5}

(ii) A ∪ B

A = {a, e, i, o, u}
B = {a, b, c}

A ∪ B = {a, b, c, e, i, o, u}

(iii) A ∪ B

A = {x : x is a natural number and a multiple of 3}
B = {x : x is a natural number less than 6}

A = {3, 6, 9, 12, …}
B = {1, 2, 3, 4, 5}

A ∪ B = {1, 2, 3, 4, 5, 6, 9, 12, …}

(iv) A ∪ B

A = {x : x is a natural number and 1 < x ≤ 6}
B = {x : x is a natural number and 6 < x < 10}

A = {2, 3, 4, 5, 6}
B = {7, 8, 9}

A ∪ B = {2, 3, 4, 5, 6, 7, 8, 9}

(v) A ∪ B

A = {1, 2, 3}
B = φ

A ∪ φ = {1, 2, 3}

🧠 Memory Hack:
Union = Combine + Remove duplicates
Union with φ changes nothing
If one set is empty, the union is the other set itself.

2. Let A = { a, b }, B = {a, b, c}. Is A ⊂ B ? What is A ∪ B ?

Solution: Let A = { a, b }, B = { a, b, c }

Since every element of A is present in B, A ⊂ B.

A ∪ B = { a, b, c }

3. If A and B are two sets such that A ⊂ B, then what is A ∪ B ?

Solution: If A ⊂ B, then find A ∪ B.

When one set is a subset of another, the union becomes the larger set.

A ∪ B = B

Brain Rule: Smaller set + Bigger set = Bigger set

4. If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8 } and D = { 7, 8, 9, 10 };
find
(i) A ∪ B (ii) A ∪ C (iii) B ∪ C (iv) B ∪ D
(v) A ∪ B ∪ C (vi) A ∪ B ∪ D (vii) B ∪ C ∪ D

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

(i) A ∪ B = {1, 2, 3, 4, 5, 6}

(ii) A ∪ C = {1, 2, 3, 4, 5, 6, 7, 8}

(iii) B ∪ C = {3, 4, 5, 6, 7, 8}

(iv) B ∪ D = {3, 4, 5, 6, 7, 8, 9, 10}

(v) A ∪ B ∪ C = {1, 2, 3, 4, 5, 6, 7, 8}

(vi) A ∪ B ∪ D = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

(vii) B ∪ C ∪ D = {3, 4, 5, 6, 7, 8, 9, 10}

Union Memory Pattern:
Union = collect all + remove repeats
Watch how numbers grow continuously across unions.

5. Find the intersection of each pair of sets of (i) X = {1, 3, 5} Y = {1, 2, 3}
(ii) A = [ a, e, i, o, u} B = {a, b, c}
(iii) A = {x : x is a natural number and multiple of 3}
B = {x : x is a natural number less than 6}
(iv) A = {x : x is a natural number and 1 < x ≤6 }
B = {x : x is a natural number and 6 < x < 10 }
(v) A = {1, 2, 3}, B = φ

Intersection Rule:
The intersection of two sets consists of only those elements which are common to both sets.

A ∩ B = { x : x ∈ A and x ∈ B }

(i) X = {1, 3, 5}, Y = {1, 2, 3}

X ∩ Y = {1, 3}

(ii) A = {a, e, i, o, u}, B = {a, b, c}

A ∩ B = {a}

(iii)
A = {x : x is a natural number and a multiple of 3}
B = {x : x is a natural number less than 6}

A = {3, 6, 9, 12, …}
B = {1, 2, 3, 4, 5}

A ∩ B = {3}

(iv)
A = {x : x is a natural number and 1 < x ≤ 6}
B = {x : x is a natural number and 6 < x < 10}

A = {2, 3, 4, 5, 6}
B = {7, 8, 9}

A ∩ B = φ

(v) A = {1, 2, 3}, B = φ

A ∩ φ = φ

Brain Shortcut:
Intersection means COMMON ONLY.
If one set is empty, the intersection is always φ.

6. If A = { 3, 5, 7, 9, 11 }, B = {7, 9, 11, 13}, C = {11, 13, 15}and D = {15, 17}; find
(i) A ∩ B (ii) B ∩ C (iii) A ∩ C ∩ D
(iv) A ∩ C (v) B ∩ D (vi) A ∩ (B ∪ C)
(vii) A ∩ D (viii) A ∩ (B ∪ D) (ix) ( A ∩ B ) ∩ ( B ∪ C )
(x) ( A ∪ D) ∩ ( B ∪ C)

6. Let A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15}, D = {15, 17}

(i) A ∩ B = {7, 9, 11}

(ii) B ∩ C = {11, 13}

(iii) A ∩ C ∩ D = φ

(iv) A ∩ C = {11}

(v) B ∩ D = φ

(vi) A ∩ (B ∪ C) = {7, 9, 11}

(vii) A ∩ D = φ

(viii) A ∩ (B ∪ D) = {7, 9, 11}

(ix) (A ∩ B) ∩ (B ∪ C) = {7, 9, 11}

(x) (A ∪ D) ∩ (B ∪ C) = {7, 9, 11, 15}

Intersection Thinking Tip:
Always solve inside brackets first, then take what is common only.
If no element matches → answer is φ.

7. If A = {x : x is a natural number }, B = {x : x is an even natural number}
C = {x : x is an odd natural number}andD = {x : x is a prime number }, find
(i) A ∩ B (ii) A ∩ C (iii) A ∩ D
(iv) B ∩ C (v) B ∩ D (vi) C ∩ D

Solution. Let A = {x : x is a natural number}, B = {x : x is an even natural number}, C = {x : x is an odd natural number}, D = {x : x is a prime number}

(i) A ∩ B = {2, 4, 6, 8, 10, …}

(ii) A ∩ C = {1, 3, 5, 7, 9, …}

(iii) A ∩ D = {2, 3, 5, 7, 11, …}

(iv) B ∩ C = φ

(v) B ∩ D = {2}

(vi) C ∩ D = {3, 5, 7, 11, …}

Concept Lock:
Even ∩ Odd = φ
Only 2 is an even prime
All other primes are odd

8. Which of the following pairs of sets are disjoint
(i) {1, 2, 3, 4} and {x : x is a natural number and 4 ≤ x ≤ 6 }
(ii) { a, e, i, o, u } and { c, d, e, f }
(iii) {x : x is an even integer } and {x : x is an odd integer}

Question 8 (i):
Are the sets {1, 2, 3, 4} and {x : x is a natural number and 4 ≤ x ≤ 6} disjoint?

First set = {1, 2, 3, 4}
Second set = {4, 5, 6}

Common element = 4
Answer: These sets are NOT disjoint.

Question 8 (ii):
Are the sets {a, e, i, o, u} and {c, d, e, f} disjoint?

First set = {a, e, i, o, u}
Second set = {c, d, e, f}

Common element = e
Answer: These sets are NOT disjoint.

Question 8 (iii):
Are the sets {x : x is an even integer} and {x : x is an odd integer} disjoint?

Even integers = {..., −4, −2, 0, 2, 4, ...}
Odd integers = {..., −3, −1, 1, 3, 5, ...}

No common element exists.
Answer: These sets are DISJOINT.

🧠 Neurological Insight:
The brain remembers concepts better when contrast is involved. Disjoint sets work on a powerful neural mechanism called binary separation — the mind instantly checks:

➜ “Is there ANY overlap or not?”

This activates the prefrontal cortex, which specializes in decision-making and logical filtering. That’s why problems involving even vs odd or vowels vs consonants are remembered faster and longer.

Memory Rule:
If your brain can quickly say “never together”, the sets are disjoint.

9. If A = {3, 6, 9, 12, 15, 18, 21}, B = { 4, 8, 12, 16, 20 },
C = { 2, 4, 6, 8, 10, 12, 14, 16 }, D = {5, 10, 15, 20 }; find
(i) A – B (ii) A – C (iii) A – D (iv) B – A
(v) C – A (vi) D – A (vii) B – C (viii) B – D
(ix) C – B (x) D – B (xi) C – D (xii) D – C

Question 9 (i): A − B

A = {3, 6, 9, 12, 15, 18, 21}
B = {4, 8, 12, 16, 20}

Removing common element (12) from A
Answer: {3, 6, 9, 15, 18, 21}

Question 9 (ii): A − C

Common elements = 6, 12
Answer: {3, 9, 15, 18, 21}

Question 9 (iii): A − D

Common elements = 15
Answer: {3, 6, 9, 12, 18, 21}

Question 9 (iv): B − A

Common element = 12
Answer: {4, 8, 16, 20}

Question 9 (v): C − A

Common elements = 6, 12
Answer: {2, 4, 8, 10, 14, 16}

Question 9 (vi): D − A

Common element = 15
Answer: {5, 10, 20}

Question 9 (vii): B − C

Common elements = 4, 8, 12, 16
Answer: {20}

Question 9 (viii): B − D

Common element = 20
Answer: {4, 8, 12, 16}

Question 9 (ix): C − B

Removing 4, 8, 12, 16
Answer: {2, 6, 10, 14}

Question 9 (x): D − B

Common element = 20
Answer: {5, 10, 15}

Question 9 (xi): C − D

Common element = 10
Answer: {2, 4, 6, 8, 12, 14, 16}

Question 9 (xii): D − C

Common element = 10
Answer: {5, 15, 20}

🧠 Neurological Insight:
Set difference problems activate the brain’s inhibitory control system, managed by the anterior cingulate cortex. The mind learns best when it performs a mental deletion task — “keep this, remove that.”

Retention Trick:
Always read A − B as 👉 “What stays in A after B is erased.”

This left-to-right elimination creates a strong memory trace, making exam recall faster and more accurate.

10. If X= { a, b, c, d } and Y = { f, b, d, g}, find
(i) X – Y (ii) Y – X (iii) X ∩ Y

Question 10 (i): X − Y

X = {a, b, c, d}
Y = {f, b, d, g}

Common elements = b, d
Answer: {a, c}

Question 10 (ii): Y − X

Common elements = b, d
Answer: {f, g}

Question 10 (iii): X ∩ Y

Elements common to both X and Y
Answer: {b, d}

🧠 Neurological Insight:
When the brain solves set difference and intersection, it performs a rapid compare–filter–retain operation, governed by the prefrontal cortex.

Smart Recall Hack:
• X − Y → “Only X survives”
• Y − X → “Only Y survives”
• X ∩ Y → “What both agree on”

This verbal tagging converts abstract symbols into language-based memory, making exam recall quicker and more accurate.

Question 11: If R is the set of real numbers and Q is the set of rational numbers, find R − Q.

R − Q = Set of all real numbers which are not rational
These are called irrational numbers.

Answer: The set of irrational numbers (e.g., √2, π, √3, etc.)

12. State whether each of the following statement is true or false. Justify your answer.
(i) { 2, 3, 4, 5 } and { 3, 6} are disjoint sets.
(ii) { a, e, i, o, u } and { a, b, c, d }are disjoint sets.
(iii) { 2, 6, 10, 14 } and { 3, 7, 11, 15} are disjoint sets.
(iv) { 2, 6, 10 } and { 3, 7, 11} are disjoint sets.

Question 12 (i): {2, 3, 4, 5} and {3, 6}

Common element = 3
Conclusion: False (Not disjoint)

Question 12 (ii): {a, e, i, o, u} and {a, b, c, d}

Common element = a
Conclusion: False (Not disjoint)

Question 12 (iii): {2, 6, 10, 14} and {3, 7, 11, 15}

No common element
Conclusion: True (Disjoint sets)

Question 12 (iv): {2, 6, 10} and {3, 7, 11}

No common element
Conclusion: True (Disjoint sets)

🧠 Neurological Insight:
The brain identifies disjoint sets using a rapid pattern mismatch mechanism. If even one common element appears, the hippocampus flags it immediately.

One-Second Brain Rule:
• Common element present → NOT disjoint
• No overlap at all → Disjoint

For R − Q, the brain performs a filtering operation: “Remove fractions and decimals that repeat or terminate → what remains are irrationals.”

This converts abstract set theory into visual filtering, improving long-term retention.

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