Class 11 Maths NCERT Solutions Chapter 1 Ex 1.3 – Subsets

(i) Every element of {2,3,4} — namely 2, 3 and 4 — also lies in {1,2,3,4,5}.
\{2,3,4\}\subset\{1,2,3,4,5\}

(ii) a is not an element of {b,c,d}.
\{a,b,c\}\not\subset\{b,c,d\}

(iii) Every student of Class XI of your school is also a student of your school.
\{x : x \text{ is a student of Class XI}\}\subset\{x : x \text{ is a student of your school}\}

(iv) Not every circle in the plane has radius 1 unit — most circles have some other radius.
\{x : x \text{ is a circle in the plane}\}\not\subset\{x : x \text{ is a circle with radius 1 unit}\}

(v) A triangle is never a rectangle.
\{x : x \text{ is a triangle}\}\not\subset\{x : x \text{ is a rectangle}\}

(vi) Every equilateral triangle is, in particular, a triangle.
\{x : x \text{ is an equilateral triangle}\}\subset\{x : x \text{ is a triangle}\}

(vii) Every even natural number is an integer.
\{x : x \text{ is an even natural number}\}\subset\{x : x \text{ is an integer}\}

(i) Both a and b are elements of {b,c,a}, so {a,b} is in fact a subset of {b,c,a}.
False.

(ii) a and e are both vowels, so {a,e} is a subset of the set of vowels.
True.

(iii) 2\in\{1,2,3\} but 2\notin\{1,3,5\}, so the subset relation fails.
False.

(iv) a is an element of {a,b,c}, so {a} is a subset of {a,b,c}.
True.

(v) The elements of {a,b,c} are a, b and c themselves — not the set {a}. So {a} is not an element of {a,b,c}, only a subset of it.
False.

(vi) The even natural numbers less than 6 are {2,4}. The divisors of 36 are 1,2,3,4,6,9,12,18,36 — both 2 and 4 divide 36.
True.

The elements of A are exactly four things: 1,\ 2,\ \{3,4\},\ 5. Note that 3 and 4 are NOT themselves elements of A — only the set {3,4} is.

(i) For {3,4} ⊂ A, both 3 and 4 would need to be elements of A individually. Neither is.
Incorrect.

(ii) {3,4} is exactly one of the four listed elements of A.
Correct.

(iii) Since {3,4} ∈ A, the set containing it as its only element, {{3,4}}, is a subset of A.
Correct.

(iv) 1 is one of the listed elements of A.
Correct.

(v) 1 is a number, not a set — the subset symbol ⊂ only relates two sets, so this comparison doesn't apply.
Incorrect.

(vi) 1, 2 and 5 are all individually elements of A.
Correct.

(vii) The elements of A are 1, 2, {3,4}, 5 — the set {1,2,5} as a whole is not one of these four elements.
Incorrect.

(viii) 3 is not an element of A (only {3,4} is), so {1,2,3} cannot be a subset of A.
Incorrect.

(ix) φ is not listed among the elements of A.
Incorrect.

(x) The empty set is a subset of every set.
Correct.

(xi) For {φ} ⊂ A, φ itself would need to be an element of A. It isn't.
Incorrect.

(i) A set with 1 element has 2^1=2 subsets: the empty set and the set itself.

(ii) A set with 2 elements has 2^2=4 subsets.

(iii) A set with 3 elements has 2^3=8 subsets.

(iv) The empty set has exactly 2^0=1 subset — itself.

(i) −4 is excluded (strict inequality), 6 is included.

(ii) Both endpoints are excluded.

(iii) 0 is included, 7 is excluded.

(iv) Both endpoints are included.

(i) A round bracket on both ends means both endpoints are excluded.

(ii) A square bracket on both ends means both endpoints are included.

(iii) 6 is excluded, 12 is included.

(iv) −23 is included, 5 is excluded.

A universal set must be broad enough to contain the given set as a subset, while still being relevant to the context.

(i) Every right triangle is a triangle, so the set of all triangles (in a plane) serves as a suitable universal set.

(ii) Similarly, every isosceles triangle is a triangle.

A valid universal set must contain A, B and C all as subsets — every element of each must appear in it.

(i) {0,1,2,3,4,5,6} contains A and B, but C needs 8, which is missing here.
Not a valid universal set.

(ii) The empty set cannot contain any non-empty set as a subset.
Not a valid universal set.

(iii) {0,1,2,...,10} contains every element of A (1,3,5), B (2,4,6) and C (0,2,4,6,8).
Valid universal set.

(iv) {1,2,...,8} contains A and B, but C needs 0, which is missing here.
Not a valid universal set.