NCERT Solutions Class 11 Mathematics Sets Exercise 1.3

Question 1:
Make correct statements by filling in the symbols ⊂ or ⊄ in the blank spaces:
(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 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}

Answer:
(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 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 in a triangle in the same plane}
(vii) {x: x is an even natural number} ⊂ {x: x is an integer}

Question 2:
Examine whether the following statements are true or false:
(i) {a, b } ⊄ {b, c, a}
(ii) {a, e} ⊂ {x : x is a vowel in the English alphabet}
(iii) {1, 2, 3} ⊂ {1, 3, 5}
(iv) {a} ⊂ {a, b, c}
(v) {a} ∈ {a, b, c}
(vi) {x : x is an even natural number less than 6} ⊂ {x : x is a natural number which divides 36}

Answer:
(i) False. Each element of {a, b} is also an element of {b, c, a}.
(ii) True. a, e are two vowels of the English alphabet.
(iii) False. 2 ∈ {1, 2, 3}; however, 2 ∉ {1, 3, 5}
(iv) True. Each element of {a} is also an element of {a, b, c}.
(v) False. The elements of {a, b, c} are a, b, c. Therefore, {a} ⊂ {a, b, c}
(vi) True. {x: x is an even natural number less than 6} = {2, 4}
{x: x is a natural number which divides 36} = {1, 2, 3, 4, 6, 9, 12, 18, 36}

Question 3:
Let A = {1, 2, {3, 4}, 5}. Which of the following statements are incorrect and why?
(i) {3, 4} ⊂ A                  (ii) {3, 4} ∈ A                 (iii) {{3, 4}} ⊂ A          (iv) 1 ∈ A               (v) 1 ⊂ A
(vi) {1, 2, 5} ⊂ A           (vii) {1, 2, 5} ∈ A          (viii) {1, 2, 3} ⊂ A        (ix) φ∈ A
(x) φ ⊂ A (xi) {φ} ⊂ A

Answer:
Given, A = {1, 2, {3, 4}, 5}
(i) The statement {3, 4} ⊂ A is incorrect because 3 ∈ {3, 4}; however, 3 ∉ A.
(ii) The statement {3, 4} ∈A is correct because {3, 4} is an element of A.
(iii) The statement {{3, 4}} ⊂ A is correct because {3, 4} ∈ {{3, 4}} and {3, 4} ∈ A.
(iv) The statement 1∈A is correct because 1 is an element of A.
(v) The statement 1⊂ A is incorrect because an element of a set can never be a subset of itself.
(vi) The statement {1, 2, 5} ⊂ A is correct because each element of {1, 2, 5} is also an element
of A.
(vii)The statement {1, 2, 5} ∈ A is incorrect because {1, 2, 5} is not an element of A.
(viii) The statement {1, 2, 3} ⊂ A is incorrect because 3 ∈ {1, 2, 3}; however, 3 ∉ A.
(ix) The statement φ ∈ A is incorrect because φ is not an element of A.
(x) The statement φ ⊂ A is correct because φ is a subset of every set.
(xi) The statement {φ} ⊂ A is incorrect because φ∈ {φ}; however, φ∈ A.

Question 4:
Write down all the subsets of the following sets
(i) {a}                          (ii) {a, b}                        (iii) {1, 2, 3}                       (iv) φ

Answer:
(i) The subsets of {a} are φ and {a}.
(ii) The subsets of {a, b} are φ, {a}, {b}, and {a, b}.
(iii) The subsets of {1, 2, 3} are φ, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3} and {1, 2, 3}
(iv) The only subset of φ is φ.

Question 5:
How many elements has P(A), if A = φ?

Answer:
We know that if A is a set with m elements i.e., n(A) = m, then n[P(A)] = 2^m.
If A = φ, then n(A) = 0
So, n[P(A)] = 2⁰ = 1
Hence, P(A) has one element.