RBSE Solutions for Class 11 Maths Chapter 1 Sets Ex 1.3

Rajasthan Board RBSE Solutions for Class 11 Maths Chapter 1 Sets Ex 1.3 Textbook Exercise Questions and Answers.

RBSE Class 11 Maths Solutions Chapter 1 Sets Ex 1.3

Question 1.
Make correct statements by filling in the symbols ⊂ of ⊄ 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}, since 2, 3, 4 all occurs in other set.

(ii) {a, b, c) ⊄ {b, c, d}, since a not occur in other set.

(iii) {x : x is a student of class XI of your school} ⊂ {x : x student of your school}
Since XIth class student is also student of school.

(iv) {x : x is a circle in then plane} ⊂ {x : x is a circle in the same plane with radius 1 unit}
Since in first set, radius of circle may exceeds 1 unit.

(v) {x : x is a triangle in a plane} ⊂ {x : x is a rectangle in the plane}
Since, triangle is not rectangle.

(vi) {x : x is an equilateral triangle in a plane} ⊂ {x; x is a triangle in the same plane}
Since an equilateral triangle also exist in other set.

(vii) {x : x is an even natural number} ⊂ {x : x is an integer}
Since set of integer also represents natural numbers.

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}
Solution:
(i) {a, b} ⊄ {b, c, a} is false. {a, b} ⊂ {b, c, a}
Since, {a, b} is subset of set {b, c,a}.

(ii) {a, e} ⊂ {x: x is a vowel in the English alphabet} is true, because a and e are also vowels of English alphabet.

(iii) {1, 2, 3} ⊂ {1, 3, 5} is false, because element 2 of first set, not occurs in second set.

(iv) {a}, ⊂ [a, b, c} is true, because element a of first set also occurs in second set.

(v) {a} ⊂ [a, b, c} is false, because {a} is a set which is not occurs in second set.

(vi) {.r : x is a natural number less than 6} ⊂ { x : x is a natural number which divides 36} {2,4} ⊂ {1, 2, 3,4, 5, 6, 9, 12, 18, 36}
It is true because 2 and 4 are even natural numbers less than 6 which divides 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
(vii) {1, 2, 5} ⊂ A
(ix) Φ ⊂ A
(xi) {Φ} ⊂ A
Answer:
(i) (3,4} c A
This statement is not correct because {3, 4} is element of set^4. Thus, {{3,4}} czA.
Hence, 1 ∈ A, 2 ∈ A, {3,4} ∈ A, 5 ∈ A all these are elements of A.

(ii) (3,4} ∈ A is true because set {3, 4} is element of A.

(iii) {{3, A}}⊂ A is true because {3, 4} is element of set A when we write {{3, 4}} then it becomes set.
Thus, {{3,4}} ⊂ A.

(iv) 1 ∈ A is true because lis element of set A.

(v) 1 ⊂ A is not true because any element cannot be subset of any set until it is written in {} bracket.

(vi) {1, 2, 5} ⊂A is true because {1, 2, 5} is set of elements 1,2,5 of A.

(vii) {1, 2, 5} ∈ A is not true because 1, 2, 5 are elements of set A. Thus, {1,2,5} will be subset of set A.

(viii) {1, 2, 3} ⊂ A is not true because 3 is not an element of set A.

(ix) Φ ∈ A is not true because f .is not elements of set A.

(x) Φ ⊂ A is true because null set Φ is subset of all sets.

(xi) {Φ} ⊂ A is not true because Φ is null set and Φ is element of set {Φ}.

Question 4.
Write down all the subsets of the following sets
(i){a}
(ii){a, b}
(iii) {1,2,3}
(iv) Φ
Answer:
(i) Φ and {a} are subsets of set {a}.
(ii) Φ, {a}, {b} and {a, b} are subsets of {a, b}.
(iii) Φ, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3} and {1, 2, 3} are subsets of set {1,2,3}.
(iv) Φ is subset of set Φ, Φ ⊂ Φ

Question 5.
How many elements has P(A), of A = Φ?
Answer:
If A = Φ then only Φ will be element of P(A), since Φ has no element, i.e.
it is a null set. A null set has only one subset, i.e., itself. Thus, Φ ⊂ Φ
Number of subsets of Φ = 1
Number of subsets of P( A) = A= 1

Question 6.
Write the followingns Intervals?
(i) {x:x ∈ R, -4 ≤ x ≤ b
(ii) {x:x ∈ R, - 12 ≤ x ≤ -10}
(iii) {x:x ∈ R, 0 ≤ x ≤ 7}
(iv) {x:x ∈ R,3 ≤ x ≤ 4}
Answer:
(i) {x: x ∈ R, - 4 ≤ x ≤ 6} = [- 4,6]

(ii) {x:x ∈ R, -12 ≤ x ≤ -10}
Here, above interval is open interval from - 12 to -10. In which - 12 and - 10 both are not included. Thus,
{x:x ∈ R, -12 ≤ x ≤ -10} = (- 12, -10)
.
(iii) {x:x ∈ R, 0 ≤ x ≤ 7} above interval is half closed interval in which 0 is included but 7 is not included.
{x :x ∈ R, 0≤ x ≤ 7} = [0,7]

(iv) {x: x ∈ R, 3 ≤ x ≤ 4} above interval is closed interval, in which 3 and 4 both are included. Thus,
{x: x ∈ R, 3 ≤ x ≤ 4} = [3, 4]

Question 7.
Write the following intervals in set-builder form :
(i) (- 3,0)
(ii) 16,12}
(iii) (6,12]
(iv) [- 23,5)
Answer:
(i) (- 3,0) = {x :x ∈ R, -3 ≤ x ≤ 0}
(ii) [6,12] = {x:x ∈ P,6 ≤ x ≤ 12}
(iii) (6,12] = {x :x ∈ R, 6 ≤ x ≤ 12}
(iv) [-23, 5) = {x :x ∈ R, -23 ≤ x ≤ 5}

Question 8.
What universal set(s) would you propose for each of the following:
(i) The set of right triangles
(ii) The set of isosceles triangles.
Answer:
(i) Set of triangles is universal set.
(ii) Set of triangles is universal set.
∵ In the set of triangles, right angled and isosceles triangles occurs.

Question 9.
Given the sets A = {1,3, 5}, B = {2,4, 6} and C = {0, 2, 4, 6, 8}, which of the following may be considered as universal set(s) for all the three sets A, B and C:
(i) {0,1,2,3,4,5,6}
(ii) Φ
(iii) {0,1,2,3,4, 5,6, 7,8,9,10}
(iv) {1,2,3, 4,5,6,7,8}
Answer:
For sets A = {1, 3, 5}, B = {2, 4, 6} and C = {0,2, 4, 6, 8}.
Universal set will be (iii) { 0,1,2, 3,4, 5,6, 7, 8,9, 10}.
Since, it contains all elements of set A, B and C.
Thus, (iii) is correct.