TOPICS

Exercise - 1.3

Sets

**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}

(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}

(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

Give that 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

(i) The subsets of {a} = {Φ, {a}}. (ii) The subsets of {a, b} = {Φ, {a}, {b}, {a, b}}. (iii) The subsets of {1, 2, 3} = {Φ, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, {1, 2, 3}} (iv) The only subset of Φ = {Φ}.

**Question-5 :-** How many elements has P(A), if A = Φ?

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

**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}

(i) {x: x ∈ R, –4 < x ≤ 6} = (–4, 6] (ii) {x: x ∈ R, –12 < x < –10} = (–12, –10) (iii) {x: x ∈ R, 0 ≤ x < 7} = [0, 7) (iv) {x: x ∈ R, 3 ≤ x ≤ 4} = [3, 4]

**Question-7 :-** Write the following intervals in set-builder form:

(i) (–3, 0)

(ii) [6, 12]

(iii) (6, 12]

(iv) [–23, 5)

(i) (–3, 0) = {x: x ∈ R, –3 < x < 0} (ii) [6, 12] = {x: x ∈ R, 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

(i) Right triangle is a type of triangle. Therefore, the set of triangles contain all types of triangles. Hence, U = {x : x is a triangle in a plane.} (ii) Isosceles triangle is a type of triangle. Therefore, the set of triangles contain all types of triangles. Hence, 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 universals 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}

Given that sets A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}. So, U = {0, 1, 2, 3, 4, 5, 6, 8} (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. Therefore 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}.

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