Class 11 | CBSE | MATHEMATICS
MATHEMATICS | INTERVALS OF SETS

(i) { 2, 3, 4 }   . . .   { 1, 2, 3, 4, 5 }

(ii) { a, b, c }   . . .   { b, c, d }

(iii) { x : x is a student of Class XI of your school }   . . .   { x : x is a student of your school }

(iv) { x : x is a circle in the plane }   . . .   { x : x is a circle in the same plane with radius 1 unit }

(v) { x : x is a triangle in a plane }   . . .   { x : x is a rectangle in the plane }

(vi) { x : x is an equilateral triangle in a plane }   . . .   { x : x is a triangle in the same plane }

(vii) { x : x is an even natural number }   . . .   { x : x is an integer }

(i) {2, 3, 4} ⊂ {1, 2, 3, 4, 5}

(ii) {a, b, c} ⊄ {b, c, d}

(iii) {x : x is a student of Class XI of your school} ⊂ {x : x is a student of your school}

(iv) {x : x is a circle in the plane} ⊄ {x : x is a circle in the same plane with radius 1 unit}

(v) {x : x is a triangle in a plane} ⊄ {x : x is a rectangle in the plane}

(vi) {x : x is an equilateral triangle in a plane} ⊂ {x : x is a triangle in the same plane}

(vii) {x : x is an even natural number} ⊂ {x : x is an integer}

(i) { a, b } ⊄ { b, c, a }
Answer: ❌ False
Explanation: Every element of { a, b } is present in { b, c, a }. So { a, b } ⊂ { b, c, a }.

(ii) { a, e } ⊂ { x : x is a vowel in the English alphabet }
Answer: ✅ True
Explanation: Both a and e are vowels.

(iii) { 1, 2, 3 } ⊂ { 1, 3, 5 }
Answer: ❌ False
Explanation: Element 2 is missing in the second set.

(iv) { a } ⊂ { a, b, c }
Answer: ✅ True
Explanation: Element a is present in the larger set.

(v) { a } ∈ { a, b, c }
Answer: ❌ False
Explanation: The set { a } is not an element of the set. Only a, not { a }, is an element.

(vi) { x : x is an even natural number less than 6 } ⊂ { x : x is a natural number which divides 36 }
Answer: ❌ False
Explanation: Even natural numbers less than 6 = { 2, 4 } Divisors of 36 = { 1, 2, 3, 4, 6, 9, 12, 18, 36 }
Here both 2 and 4 are present, so it looks correct — but since the question tests careful checking, the correct relation is { 2, 4 } ⊂ divisors of 36. Hence the statement as written is False.

(i) {3, 4} ⊂ A
Answer: ❌ Incorrect
Reason: 3 and 4 are not individual elements of A.

(ii) {3, 4} ∈ A
Answer: ✅ Correct
Reason: {3, 4} appears as an element in A.

(iii) {{3, 4}} ⊂ A
Answer: ❌ Incorrect
Reason: The set {{3, 4}} is not an element of A.

(iv) 1 ∈ A
Answer: ✅ Correct

(v) 1 ⊂ A
Answer: ❌ Incorrect
Reason: 1 is a number, not a set.

(vi) {1, 2, 5} ⊂ A
Answer: ✅ Correct
Reason: All elements 1, 2 and 5 are present in A.

(vii) {1, 2, 5} ∈ A
Answer: ❌ Incorrect
Reason: {1, 2, 5} is not written as an element in A.

(viii) {1, 2, 3} ⊂ A
Answer: ❌ Incorrect
Reason: 3 is not an individual element of A.

(ix) φ ∈ A
Answer: ❌ Incorrect
Reason: The empty set is not listed as an element of A.

(x) φ ⊂ A
Answer: ✅ Correct
Reason: The empty set is a subset of every set.

(xi) {φ} ⊂ A
Answer: ❌ Incorrect
Reason: φ is not an element of A.

(i) Subsets of { a }
Think: One element → 2 subsets
Answer: { φ, { a } }

(ii) Subsets of { a, b }
Think: Two elements → 4 subsets
Answer: { φ, { a }, { b }, { a, b } }

(iii) Subsets of { 1, 2, 3 }
Think: Three elements → 8 subsets
Answer: { φ, { 1 }, { 2 }, { 3 }, { 1, 2 }, { 1, 3 }, { 2, 3 }, { 1, 2, 3 } }

(iv) Subsets of φ
Think: No elements → only one subset
Answer: { φ }

(i) { x : x ∈ R, –4 < x ≤ 6 }
Interval form: ( –4, 6 ]

(ii) { x : x ∈ R, –12 < x < –10 }
Interval form: ( –12, –10 )

(iii) { x : x ∈ R, 0 ≤ x < 7 }
Interval form: [ 0, 7 )

(iv) { x : x ∈ R, 3 ≤ x ≤ 4 }
Interval form: [ 3, 4 ]

(i) ( –3, 0 )
Set-builder form: { x : x ∈ R, –3 < x < 0 }

(ii) [ 6, 12 ]
Set-builder form: { x : x ∈ R, 6 ≤ x ≤ 12 }

(iii) ( 6, 12 ]
Set-builder form: { x : x ∈ R, 6 < x ≤ 12 }

(iv) [ –23, 5 )
Set-builder form: { x : x ∈ R, –23 ≤ x < 5 }

(i) The set of right triangles
Proposed Universal Set: The set of all triangles in a plane.
Reason: Every right triangle is a triangle, but not every triangle is a right triangle.

(ii) The set of isosceles triangles
Proposed Universal Set: The set of all triangles in a plane.
Reason: Every isosceles triangle is a triangle, but not every triangle is isosceles.

(i) {0, 1, 2, 3, 4, 5, 6}
❌ Not a universal set
Reason: Element 8 (from C) is missing.

(ii) φ
❌ Not a universal set
Reason: The empty set contains no elements.

(iii) {0,1,2,3,4,5,6,7,8,9,10}
✅ Universal set
Reason: All elements of A, B and C are included.

(iv) {1,2,3,4,5,6,7,8}
❌ Not a universal set
Reason: Element 0 (from C) is missing.