Chapter 1 – Sets (Ex – 1.3)

Sets (Ex – 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}

Solution.
(i) {2, 3, 4} ⊂ {11, 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}

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} ∈ la, b, c}
(vi) {x: x is an even natural number less than 6} ⊂ {x: x is a natural number which divides 36}
Solution.

Question 3.
Let A = {1, 2, {3, 4}, 5}. Which of the following statements are incorrect and why?

Solution.

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

Solution.
(i) Number of elements in given set = 1
Number of subsets of given set = 2¹ = 2
∴ Subsets of given set are φ , {a}.

(ii) Number of elements in given set = 2
Number of subsets of given set = 2¹² = 4
∴ Subsets of given set are φ, {a}, {b}, {a, b}.

(iii) Number of elements in given set = 3
Number of subsets of given set = 2³ = 8
Subsets of given set are φ, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, {1, 2, 3}.

(iv) Number of elements in given set = 0
Number of subsets of given set = 2⁰= 1
∴ Subset of given set is φ.

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

Solution.
Number of elements in set A = 0
Number of subset of set A = 2⁰ = 1
Hence, number of elements of P(A) is 1.

Question 6.
Write the following as intervals:
(i) {x: x ∈ R, -4 < x ≤ 6}
(ii) {x: x ∈ R, -12 < x < -10}
(iii) {x: x ∈ R, 0 ≤ x < 7}
(iv) {x: x ∈ R, 3 ≤ x ≤ 4}

Solution.
(i)Let A = {x: x ∈ R, -4 < x ≤ 6}
It can be written in the form of interval as (-4, 6)
(ii) Let A= {x: x ∈ R, -12 < x < -10}
It can be written in the form of interval as (-12, -10)
(iii) Let A = {x: x ∈ R, 0 ≤ x < 7}
It can be written in the form of interval as (0, 7).
(iv) Let A = {x: x ∈ R, 3 ≤ x ≤ 4}
It can be written in the form of interval as (3,4).

Question 7.
Write the following intervals in set-builder form:
(i) (-3,0)
(ii) [6, 12]
(iii) (6, 12]
(iv) [-23, 5)

Solution.
(i) The interval (-3, 0) can be written in set-builder form as {x : x ∈ R,-3 < x < 0}.
(ii) The interval [6, 12] can be written in set-builder form as {x : x ∈ R, 6 ≤ x ≤ 12}.
(iii) The interval (6, 12] can be written in set-builder form as {x : x ∈ R, 6 < x ≤ 12}
(iv) The interval [-23,5) can be written in set-builder form as {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.

Solution.
(i) Right triangle is a type of triangle. So the set of triangles contain all types of triangles.
∴ U = {x : x is a triangle in a plane}

(ii) Isosceles triangle is a type of triangle. So the set of triangles contain all types of triangles.
∴ U = }x : x is a triangle in a plane}

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}

Solution.

(i) {0, 1, 2, 3, 4, 5, 6} is not a universal set for A, B, C because 8 ∈ C but 8 is not a member of {0, 1, 2, 3, 4, 5, 6}.
(ii) φ is a set which contains no element. So it is not a universal set for A, B, C.
(iii) {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} is a universal set for A, B, C because all members of A, B, C are present in {0,1 , 2, 3, 4, 5, 6, 7, 8, 9, 10).
(iv) (1, 2, 3, 4, 5, 6, 7, 8) is not a universal set for A, B, C because 0 ∈ C but 0 is not a member of {1, 2, 3, 4, 5, 6, 7, 8)